#236215

order	512
nilpotency class	3
isoclinism	isoclinic class of #233958
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^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=dhi,e^c=ei,f^c=fi

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, 143411);