#509654

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3155	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416850);