#6643189

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37296	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fh,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efh,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei

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, 6550385);