#1301621

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

2012 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)(5,6,8,10)(7,11,12,9)(21,22)(23,28)(24,27)(25,29)(26,31)(30,34)(32,35)(33,36);(13,14,16,18)(15,17,19,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24,25,32)(22,27,29,35)(23,30,31,36)(26,33,28,34);(5,7,8,12)(6,9,10,11)(21,23)(22,28)(24,30)(25,31)(26,29)(27,34)(32,36)(33,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,16,19)(14,17,18,20)(21,22)(23,26)(24,27)(25,29)(28,31)(30,33)(32,35)(34,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=i,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, 1208817);