#476366

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1093 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383562);