#397322

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

1063 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)(8,9)(11,12)(13,14)(15,26)(16,22)(17,33)(18,24)(19,25)(20,36)(21,29)(23,34)(27,38)(28,41)(30,40)(31,44)(32,42)(35,39)(37,43);(1,3,4,2)(6,9)(10,12)(13,15)(14,21)(16,31)(17,28)(18,29)(19,35)(20,30)(22,41)(23,38)(24,26)(25,40)(27,32)(33,44)(34,43)(36,39)(37,42);(2,3)(13,16,18,17)(14,22,24,33)(15,27,29,43)(19,34,20,32)(21,37,26,38)(23,36,42,25)(28,35,31,30)(39,44,40,41);(5,7,8,11)(6,10,9,12)(13,17,18,16)(14,23,24,42)(15,28,29,31)(19,34,20,32)(21,38,26,37)(22,36,33,25)(27,35,43,30)(39,44,40,41)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,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,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, 304518);