#504425

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

569 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)(3,6)(4,5)(7,8)(9,10,13,20,14,21,16,23)(11,24,27,43,28,45,30,17)(12,31,32,48,33,18,35,19)(15,22,36,50,37,51,40,53)(25,57,59,41,60,54,63,71)(26,42,61,46,62,72,64,55)(29,56,65,70,66,52,38,44)(34,67,39,69,58,47,68,49);(9,11,14,28)(10,17,21,43)(12,26,33,62)(13,30,16,27)(15,38,37,65)(18,46,31,55)(19,42,48,72)(20,45,23,24)(22,52,51,56)(25,58,60,34)(29,36,66,40)(32,64,35,61)(39,63,68,59)(41,67,71,47)(44,50,70,53)(49,57,69,54);(2,5)(6,8)(9,12)(10,18)(11,25)(13,32)(14,33)(15,39)(16,35)(17,41)(19,20)(21,31)(22,49)(23,48)(24,54)(26,38)(27,59)(28,60)(29,61)(30,63)(34,40)(36,58)(37,68)(42,70)(43,71)(44,72)(45,57)(46,52)(47,53)(50,67)(51,69)(55,56)(62,65)(64,66);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,26)(13,35)(14,33)(15,34)(16,32)(17,42)(18,20)(21,48)(22,49)(23,31)(24,55)(25,38)(27,64)(28,62)(29,63)(30,61)(36,68)(37,58)(39,40)(41,70)(43,72)(44,71)(45,46)(47,50)(51,69)(52,57)(53,67)(54,56)(59,66)(60,65)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411621);