#416993

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

838 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:

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

Polycyclic presentation:

2^9|a^2=efg,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, 324189);