#355003

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×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

1226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(2,4)(5,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,25,23,26,22,20)(19,24,29,32,27,31,30,28);(10,14)(15,16)(17,19,22,30,23,27,21,29)(18,24,20,28,26,31,25,32);(1,2,3,4)(9,11,13,12)(10,15,14,16)(19,27)(24,31)(28,32)(29,30);(9,12,13,11)(10,16,14,15)(17,20,22,26,23,25,21,18)(19,28,30,31,27,32,29,24)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=eh,g^2=i,h^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, 262199);