#393874

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1162 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)(13,14,19,23,17,21,18,31)(15,20,26,36,25,33,32,35)(16,27,30,37,28,41,29,22)(24,38,40,34,39,44,42,43);(1,2,3,4)(6,9)(10,11)(13,15,18,26,17,25,19,32)(14,20,23,35,21,33,31,36)(16,24,29,40,28,39,30,42)(22,34,27,38,37,43,41,44);(5,7,8,12)(6,10,9,11)(14,21)(20,33)(22,37)(23,31)(27,41)(34,43)(35,36)(38,44);(13,16,18,29,17,28,19,30)(14,22,23,27,21,37,31,41)(15,24,26,40,25,39,32,42)(20,34,35,38,33,43,36,44)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=fi,d^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,e^b=ei,h^b=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301070);