#6683015

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

2100 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)(15,20)(16,24)(17,21)(18,23)(19,22)(25,26)(27,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,19,25)(14,20,22,26)(16,21,27,23)(17,28,18,24);(2,4)(13,16,17,26,19,27,18,20)(14,15,23,28,22,25,21,24);(6,10)(9,11)(13,17,19,18)(14,21,22,23)(15,24,25,28)(16,26,27,20);(2,4)(7,12)(9,11)(13,17,19,18)(14,21,22,23)(15,24,25,28)(16,26,27,20)

Polycyclic presentation:

2^9|b^2=fi,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

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, 6590211);