#416362

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323558);