#416609

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
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×#3153	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1730 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)(15,25)(16,30)(17,24)(18,23)(19,22)(20,27)(21,33)(26,41)(28,36)(29,37)(31,38)(32,39)(34,40)(35,43)(42,44);(1,2,3,4)(6,9)(10,11)(13,15,18,28,19,29,17,27)(14,20,24,37,22,36,23,25)(16,31,33,43,34,42,32,26)(21,38,30,41,39,44,40,35);(13,16,17,33,19,34,18,32)(14,21,23,30,22,39,24,40)(15,26,27,31,29,43,28,42)(20,35,25,38,36,41,37,44);(5,7,8,12)(6,10,9,11)(14,22)(20,36)(21,39)(23,24)(25,37)(30,40)(35,41)(38,44)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=hi,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,e^c=ei,g^c=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, 323805);