#431962

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1170 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)(5,6,8,10)(7,11,12,9)(13,14)(15,24)(16,30)(17,21)(18,22)(19,27)(20,33)(23,34)(25,29)(26,40)(28,37)(31,38)(32,36)(35,43)(39,42)(41,44);(2,3)(5,7,8,12)(6,9,10,11)(13,14)(15,25)(16,20)(17,21)(18,23)(19,28)(22,34)(24,29)(26,40)(27,37)(30,33)(31,36)(32,38)(35,42)(39,43)(41,44);(1,2,4,3)(6,10)(9,11)(13,15)(14,19)(16,26)(17,27)(18,29)(20,35)(21,24)(22,37)(23,25)(28,34)(30,44)(31,41)(32,43)(33,42)(36,39)(38,40);(1,3,4,2)(7,12)(9,11)(13,16,17,31)(14,20,21,36)(15,26,27,41)(18,32,34,33)(19,35,24,39)(22,30,23,38)(25,40,37,44)(28,42,29,43)

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=efi,d^2=ef,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, 339158);