#236177

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1414 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143373);