#766886

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=hi,c^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 674082);