#393593

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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

894 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:

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

Polycyclic presentation:

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

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, 300789);