#405918

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1226 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)(7,9)(8,10)(11,12)(13,14,17,32,19,23,18,22)(15,20,25,36,27,37,26,35)(16,28,33,21,31,39,30,40)(24,41,29,34,43,38,42,44);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15,19,27)(14,20,23,37)(16,29,31,42)(17,25,18,26)(21,38,40,41)(22,35,32,36)(24,33,43,30)(28,34,39,44);(1,3,4,2)(13,16)(14,21)(15,24)(17,30)(18,33)(19,31)(20,34)(22,39)(23,40)(25,42)(26,29)(27,43)(28,32)(35,38)(36,41)(37,44);(6,10)(9,12)(13,17,19,18)(14,22,23,32)(15,25,27,26)(16,30,31,33)(20,35,37,36)(21,39,40,28)(24,42,43,29)(34,38,44,41)

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,f^c=fi,g^c=gi

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, 313114);