#354694

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₈	C₂³
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261890);