#416062

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323258);