#6810946

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(6,10)(7,8)(13,14,17,20)(15,22,21,18)(16,23,24,28)(19,26,27,25);(2,3)(5,6)(7,12)(8,11)(9,10)(13,15)(14,18)(16,23)(17,21)(19,26)(20,22)(24,28)(25,27);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,19)(15,23)(17,24)(18,26)(20,27)(21,28)(22,25);(5,8,9,7)(6,11,10,12)(13,14)(15,22)(16,19)(17,20)(18,21)(23,25)(24,27)(26,28);(1,3,4,2)(14,20)(15,21)(19,27)(23,28)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718142);