#6682349

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682340
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	#36097	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2448 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)(15,22,25,28)(16,20,17,26)(19,23,27,24);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(2,4)(13,15,16,23,18,25,17,24)(14,19,26,28,21,27,20,22);(6,10)(9,11)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28);(2,4)(7,12)(9,11)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589545);