#398209

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1314 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=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, 305405);