#504426

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

Polycyclic presentation:

2^9|a^2=ei,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, 411622);