#7624868

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624842
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#62330	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1271 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2)(3,7)(4,6)(5,8)(9,10,16,21)(11,17,25,42)(12,26,32,57)(13,19,15,37)(14,35,36,64)(18,43,46,30)(20,34,47,28)(22,48,27,54)(23,39,33,41)(24,52,53,67)(29,55,31,63)(38,60,44,51)(40,58,68,49)(45,61,56,62)(50,69,59,70)(65,72,66,71);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,27)(13,33)(14,24)(15,23)(16,25)(18,44)(19,41)(20,40)(21,42)(22,32)(26,54)(28,49)(29,50)(30,60)(31,59)(34,58)(35,52)(36,53)(37,39)(38,46)(43,51)(45,65)(47,68)(48,57)(55,69)(56,66)(61,72)(62,71)(63,70)(64,67);(9,12,28,61,15,31,36,43,16,32,34,62,13,29,14,30)(10,18,35,55,19,45,47,57,21,46,64,63,37,56,20,26)(11,22,49,71,23,50,53,60,25,27,58,72,33,59,24,51)(17,38,52,70,41,66,68,54,42,44,67,69,39,65,40,48);(3,8)(5,7)(9,11)(10,17)(12,22)(13,33)(14,24)(15,23)(16,25)(18,38)(19,41)(20,40)(21,42)(26,48)(27,32)(28,49)(29,59)(30,51)(31,50)(34,58)(35,52)(36,53)(37,39)(43,60)(44,46)(45,66)(47,68)(54,57)(55,70)(56,65)(61,71)(62,72)(63,69)(64,67);(2,6)(5,7)(9,13,16,15)(10,19,21,37)(11,23,25,33)(12,29,32,31)(14,34,36,28)(17,39,42,41)(18,45,46,56)(20,35,47,64)(22,50,27,59)(24,49,53,58)(26,55,57,63)(30,62,43,61)(38,65,44,66)(40,67,68,52)(48,69,54,70)(51,71,60,72)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532064);