#406458

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1328 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)(7,10)(8,9)(11,12)(13,14)(15,21)(16,30)(17,24)(18,23)(19,25)(20,27)(22,32)(26,34)(28,36)(29,35)(31,39)(33,38)(37,41)(40,43)(42,44);(6,9)(10,12)(13,15,18,28,19,29,17,27)(14,20,24,35,25,36,23,21)(16,26,32,41,33,42,31,40)(22,34,30,43,39,44,38,37);(1,2,3,4)(13,15,19,29)(14,21,25,35)(16,26,33,42)(17,27,18,28)(20,23,36,24)(22,37,39,43)(30,34,38,44)(31,40,32,41);(5,7,8,11)(6,10,9,12)(13,16,18,32,19,33,17,31)(14,22,24,30,25,39,23,38)(15,26,28,41,29,42,27,40)(20,34,35,43,36,44,21,37)

Polycyclic presentation:

2^9|b^2=h,c^2=fi,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,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, 313654);