#429768

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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)(7,10)(8,12)(9,11)(13,14)(15,20)(16,29)(17,23)(18,24)(19,34)(21,30)(22,31)(25,36)(26,44)(27,37)(28,38)(32,39)(33,41)(35,42)(40,43);(2,3)(5,7,9,8)(6,10,11,12)(13,15,17,27)(14,20,23,37)(16,26,30,43)(18,32,19,28)(21,40,29,44)(22,36,41,35)(24,39,34,38)(25,33,42,31);(1,2,4,3)(6,11)(10,12)(13,16)(14,21)(15,25)(17,30)(18,33)(19,31)(20,35)(22,24)(23,29)(26,32)(27,42)(28,43)(34,41)(36,37)(38,44)(39,40);(1,3,4,2)(5,8,9,7)(6,12,11,10)(13,16)(14,22)(15,26)(17,30)(18,31)(19,33)(20,36)(21,24)(23,41)(25,32)(27,43)(28,42)(29,34)(35,37)(38,44)(39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336964);