#416644

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

1756 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)(13,14,19,24)(15,20,26,35)(16,29,31,40)(17,23,18,22)(21,33,39,32)(25,38,41,44)(27,37,28,36)(30,34,42,43);(1,2,3,4)(6,9)(10,11)(13,15)(14,20)(16,30,31,42)(17,27)(18,28)(19,26)(21,38,39,44)(22,36)(23,37)(24,35)(25,32,41,33)(29,43,40,34);(13,16,17,32,19,31,18,33)(14,21,22,29,24,39,23,40)(15,25,27,30,26,41,28,42)(20,34,36,38,35,43,37,44);(5,7,8,12)(6,10,9,11)(15,26)(20,35)(25,41)(27,28)(30,42)(34,43)(36,37)(38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,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, 323840);