#493258

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₂³
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,11)(8,12)(9,10)(13,14,16,21,17,22,26,34,18,19,27,30,28,31,38,41)(15,20,23,32,24,33,35,42,25,29,36,39,37,40,43,44);(1,2,4,3)(6,10)(11,12);(1,3,4,2)(5,7,9,8)(6,11,10,12)(14,19)(20,29)(21,30)(22,31)(32,39)(33,40)(34,41)(42,44);(5,8,9,7)(6,12,10,11)(13,15,16,23,17,24,26,35,18,25,27,36,28,37,38,43)(14,20,21,32,22,33,34,42,19,29,30,39,31,40,41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400454);