#262265

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×#562	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

897 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)(8,17)(9,16)(10,14)(11,13)(12,18)(19,20)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,36)(34,35);(1,2,4,3)(5,7)(6,8)(9,12)(10,17)(11,20)(13,18)(14,15)(16,19)(22,26)(25,32)(29,36)(31,34);(5,8,10,15)(6,12,14,20)(7,16,17,13)(9,19,11,18)(21,23,24,30)(22,27,26,35)(25,34,32,31)(28,36,33,29);(1,3,4,2)(5,9,10,11)(6,13,14,16)(7,12,17,20)(8,18,15,19)(22,26)(23,30)(28,33)(31,34)

Polycyclic presentation:

2^9|b^2=eh,c^2=h,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, 169461);