#399269

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

995 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)(7,10)(8,9)(11,12)(13,14)(15,25)(16,31)(17,22)(18,23)(19,28)(20,32)(21,34)(24,35)(26,36)(27,38)(29,40)(30,37)(33,39)(41,42)(43,44);(1,2,4,3)(6,9)(10,12)(13,15)(14,19)(16,27)(17,28)(18,30)(20,38)(21,36)(22,25)(23,40)(24,37)(26,33)(29,35)(31,44)(32,43)(34,42)(39,41);(1,2,4,3)(13,16)(14,20)(15,26)(17,32)(18,34)(19,36)(21,24)(22,31)(23,39)(25,41)(27,30)(28,42)(29,43)(33,35)(37,44)(38,40);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,27)(17,32)(18,33)(19,36)(20,24)(22,39)(23,31)(25,41)(26,30)(28,43)(29,42)(34,35)(37,38)(40,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=ei,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,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306465);