#6683007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,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

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, 6590203);