#396089

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1395 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 303285);