#416979

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

836 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)(5,6,9,10)(7,11,8,12)(13,14)(15,23)(16,20)(17,33)(18,21)(19,26)(22,24)(25,41)(27,35)(28,36)(29,44)(30,40)(31,39)(32,38)(34,43)(37,42);(6,10)(11,12)(13,15,24,27,18,28,17,26)(14,19,33,36,21,35,22,23)(16,29,31,25,32,43,30,42)(20,37,40,34,38,41,39,44);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,30,18,32,24,31)(14,20,22,39,21,38,33,40)(15,25,26,29,28,42,27,43)(19,34,23,37,35,44,36,41);(5,8,9,7)(6,11,10,12)(14,21)(19,35)(20,38)(22,33)(23,36)(34,44)(37,41)(39,40)

Polycyclic presentation:

2^9|a^2=efg,b^2=eh,c^2=e,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 324175);