#262181

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

900 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169377);