#495328

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402524);