#412906

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

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=e,d^2=hi,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, 320102);