#6683035

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	C₂×#3061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2028 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590231);