#417184

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

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, 324380);