#416611

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1684 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)(13,14,19,24)(15,25,27,37)(16,30,34,39)(17,23,18,22)(20,29,36,28)(21,32,40,33)(26,41,42,35)(31,38,43,44);(1,2,3,4)(6,9)(10,11)(13,15,17,28,19,27,18,29)(14,20,22,25,24,36,23,37)(16,31,32,26,34,43,33,42)(21,38,39,35,40,44,30,41);(13,16,17,33,19,34,18,32)(14,21,22,30,24,40,23,39)(15,26,28,31,27,42,29,43)(20,35,25,38,36,41,37,44);(5,7,8,12)(6,10,9,11)(15,27)(20,36)(25,37)(26,42)(28,29)(31,43)(35,41)(38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,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,d^b=di,e^c=ei,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, 323807);