#457876

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
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

913 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365072);