#318480

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=egi,c^2=i,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dhi,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, 225676);