#1298350

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

1956 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)(23,29)(24,30)(25,33)(26,27)(28,35)(31,32)(34,36);(7,12)(9,11)(13,14,17,16)(15,19,20,18)(22,27)(23,30)(28,33)(31,36);(5,7,8,12)(6,9,10,11)(13,15,17,20)(14,18,16,19)(21,23)(22,24)(25,31)(26,30)(27,29)(28,32)(33,34)(35,36);(1,2,3,4)(21,24)(22,23)(25,34)(26,29)(27,30)(28,36)(31,33)(32,35);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,19,20,18)(21,25,26,35)(22,28,27,33)(23,31,30,36)(24,32,29,34)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fh,d^2=fi,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, 1205546);