#1263494

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2258 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)(5,6,10,12)(7,8,11,9)(13,14,17,20)(15,19,16,18)(21,22,26,27)(23,24,29,32)(25,28,34,35)(30,31,36,33);(5,7,10,11)(6,8,12,9)(13,14,17,20)(15,18,16,19)(22,27)(23,29)(28,35)(30,36);(5,8,10,9)(6,7,12,11)(13,14)(15,18)(16,19)(17,20)(21,23)(22,24)(25,30)(26,29)(27,32)(28,31)(33,35)(34,36);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15,17,16)(14,18,20,19)(21,24)(22,23)(25,33)(26,32)(27,29)(28,36)(30,35)(31,34);(5,9,10,8)(6,11,12,7)(13,16,17,15)(14,19,20,18)(21,25)(22,28)(23,30)(24,31)(26,34)(27,35)(29,36)(32,33)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fgh,d^2=fh,e^2=g,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, 1170690);