#504428

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×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

578 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)(9,10,16,38,15,22,13,20)(11,23,33,17,28,49,26,43)(12,18,31,19,32,47,29,45)(14,21,39,65,37,52,34,50)(24,53,56,40,57,67,55,68)(25,41,59,44,60,70,58,69)(27,51,63,42,61,72,35,64)(30,46,36,48,54,66,62,71);(9,11,15,28)(10,17,22,43)(12,25,32,60)(13,33,16,26)(14,35,37,63)(18,44,47,69)(19,41,45,70)(20,49,38,23)(21,51,52,72)(24,54,57,30)(27,39,61,34)(29,59,31,58)(36,55,62,56)(40,66,68,46)(42,65,64,50)(48,53,71,67);(2,5)(6,8)(9,12)(10,18)(11,24)(13,29)(14,36)(15,32)(16,31)(17,40)(19,38)(20,45)(21,48)(22,47)(23,53)(25,35)(26,55)(27,59)(28,57)(30,34)(33,56)(37,62)(39,54)(41,64)(42,70)(43,68)(44,51)(46,50)(49,67)(52,71)(58,61)(60,63)(65,66)(69,72);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,25)(13,31)(14,30)(15,32)(16,29)(17,41)(18,38)(20,47)(21,48)(22,45)(23,44)(24,35)(26,59)(27,55)(28,60)(33,58)(34,36)(37,54)(39,62)(40,64)(42,68)(43,70)(46,65)(49,69)(50,66)(51,53)(52,71)(56,61)(57,63)(67,72)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,d^2=f,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, 411624);