#366108

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1008 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)(7,11)(8,9)(10,12)(13,14,18,23,19,24,17,22)(15,20,26,34,27,35,31,39)(16,21,30,38,29,37,32,40)(25,33,42,44,41,43,28,36);(2,4)(6,9)(10,11)(13,15,18,26,19,27,17,31)(14,20,23,34,24,35,22,39)(16,28,30,25,29,42,32,41)(21,36,38,33,37,44,40,43);(2,4)(13,16)(14,21)(15,25)(17,32)(18,30)(19,29)(20,33)(22,40)(23,38)(24,37)(26,42)(27,41)(28,31)(34,44)(35,43)(36,39);(2,4)(5,7,8,12)(6,10,9,11)(16,29)(21,37)(25,41)(28,42)(30,32)(33,43)(36,44)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273304);