#1297296

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1895 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1204492);