#7623133

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
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	#33524	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1492 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,26)(13,17)(14,19)(15,35)(16,38)(18,29)(20,37)(21,34)(23,51)(24,54)(25,42)(27,58)(28,60)(30,64)(31,65)(32,50)(33,59)(36,47)(39,46)(40,61)(41,67)(43,62)(44,68)(45,63)(48,52)(49,56)(53,70)(55,69)(57,72)(66,71);(2,5)(6,8)(9,11,14,13)(10,17,19,22)(12,23,28,27)(15,24,33,31)(16,25,34,32)(18,40,43,48)(20,41,45,49)(21,42,38,50)(26,58,60,51)(29,52,62,61)(30,53,39,57)(35,65,59,54)(36,55,68,66)(37,56,63,67)(44,69,47,71)(46,70,64,72);(9,12,14,28)(10,18,19,43)(11,23,13,27)(15,36,33,68)(16,39,34,30)(17,40,22,48)(20,46,45,64)(21,47,38,44)(24,55,31,66)(25,57,32,53)(26,59,60,35)(29,63,62,37)(41,70,49,72)(42,71,50,69)(51,65,58,54)(52,67,61,56);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,17,19,22)(12,27,28,23)(15,31,33,24)(16,32,34,25)(18,40,43,48)(20,41,45,49)(21,42,38,50)(26,58,60,51)(29,61,62,52)(30,57,39,53)(35,65,59,54)(36,66,68,55)(37,67,63,56)(44,69,47,71)(46,70,64,72);(11,13)(17,22)(23,27)(24,31)(25,32)(40,48)(41,49)(42,50)(51,58)(52,61)(53,57)(54,65)(55,66)(56,67)(69,71)(70,72)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530329);