#354769

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1214 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=fg,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261965);