#496158

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fh,d^2=eg,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,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, 403354);