#6474920

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,30,42)(20,25,33,37)(22,35,28,26)(24,38,36,31)(29,43,32,46)(34,48,44,41)(39,47,40,45);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(24,36)(25,37)(26,35)(27,42)(39,40)(41,48)(43,46);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,19,28,38,21,30,22,31)(18,24,35,42,23,36,26,27)(20,32,44,45,33,29,34,47)(25,40,48,43,37,39,41,46);(11,15)(13,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,39)(26,41)(27,43)(28,44)(30,32)(31,45)(35,48)(36,40)(38,47)(42,46);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(20,33)(25,37)(29,32)(34,44)(39,40)(41,48)(43,46)(45,47)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6382116);