#410068

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1112 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317264);