#351872

order	512
nilpotency class	3
isoclinism	isoclinic class of #347758
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	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

757 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ci,g^d=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259068);