#496152

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

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

520 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)(16,24)(17,27)(19,29)(21,30)(22,34)(25,39)(26,37)(28,38)(31,40)(32,44)(33,43)(35,36)(41,42);(5,7,9,8)(6,10,11,12)(13,14,18,20)(15,19,26,31)(16,21,28,33)(17,22,27,34)(23,29,37,40)(24,30,38,43)(25,32,35,41)(36,42,39,44);(13,15,24,36,17,25,28,37,18,26,38,39,27,35,16,23)(14,19,30,42,22,32,33,40,20,31,43,44,34,41,21,29);(1,3,4,2)(5,8,9,7)(6,10,11,12)(14,20)(19,31)(21,33)(22,34)(29,40)(30,43)(32,41)(42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fh,d^2=eg,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,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, 403348);