#417128

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1384 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fg,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=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, 324324);