#494086

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401282);