#262260

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
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

889 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)(5,6,9,14)(7,15,18,11)(8,12,10,13)(16,20,17,19)(21,22,26,30)(23,27,32,36)(24,29,25,28)(31,34,33,35);(1,2,4,3)(5,7,9,18)(6,11,14,15)(8,17,10,16)(12,20,13,19)(21,23)(22,27)(24,31)(25,33)(26,32)(28,35)(29,34)(30,36);(6,12)(7,16)(8,10)(13,14)(17,18)(19,20)(22,28)(24,25)(27,34)(29,30)(31,33)(35,36);(1,3,4,2)(5,8,9,10)(6,13,14,12)(7,17,18,16)(11,19,15,20)(21,24,26,25)(22,29,30,28)(23,31,32,33)(27,34,36,35)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ei,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,d^c=dh,f^c=fh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169456);