#234015

order	512
nilpotency class	3
isoclinism	isoclinic class of #234006
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1389 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,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^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, 141211);