#411749

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2142 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)(10,12)(13,14,18,22)(15,23,27,37)(16,20,28,34)(17,30,24,21)(19,31,33,26)(25,36,39,44)(29,41,38,35)(32,42,43,40);(5,6)(7,8)(13,15,24,31,18,27,17,26)(14,19,30,37,22,33,21,23)(16,25,38,42,28,39,29,40)(20,32,41,44,34,43,35,36);(2,4)(6,8)(13,16)(14,20)(15,25)(17,29)(18,28)(19,32)(21,35)(22,34)(23,36)(24,38)(26,40)(27,39)(30,41)(31,42)(33,43)(37,44);(9,10,11,12)(16,28)(20,34)(25,39)(29,38)(32,43)(35,41)(36,44)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318945);