#496056

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

610 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ei,d^2=g,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,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, 403252);