#412903

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

836 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 320099);