#262513

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

946 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)(8,14)(9,13)(10,12)(11,17)(16,20)(18,19)(21,22,26,30)(23,31,33,35)(24,29,25,28)(27,32,36,34);(1,2,4,3)(5,7)(6,11)(8,16)(9,17)(10,18)(12,20)(13,15)(14,19)(23,32)(24,25)(27,35)(28,29)(31,36)(33,34);(6,12)(8,10)(11,19)(13,14)(15,20)(16,18)(21,23,26,33)(22,27,30,36)(24,32,25,34)(28,35,29,31);(1,3,4,2)(5,8,9,10)(6,12,13,14)(7,16,17,18)(11,20,15,19)(23,33)(27,36)(31,35)(32,34)

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=i,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,f^c=fh,g^c=gi

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, 169709);