#262239

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45996	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1123 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=hi,d^2=e,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, 169435);