#1275935

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

2038 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)(15,19)(17,20)(21,22)(23,27)(24,28)(25,26)(29,35)(30,33)(31,36)(32,34);(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,17,19,20)(22,26)(27,33)(28,34)(35,36);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15)(14,17)(16,19)(18,20)(21,22)(23,27)(24,28)(25,26)(29,35)(30,33)(31,36)(32,34);(13,15)(14,17)(16,19)(18,20)(21,23,25,30)(22,27,26,33)(24,31,32,29)(28,36,34,35);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,15)(14,17)(16,19)(18,20)(21,24,25,32)(22,28,26,34)(23,29,30,31)(27,35,33,36)

Polycyclic presentation:

2^9|b^2=fhi,c^2=fh,d^2=f,e^2=h,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, 1183131);