#1297734

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

1893 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)(6,9)(7,11)(21,22,27,31)(23,29,25,28)(24,34,30,26)(32,36,35,33);(13,14,16,18)(15,17,19,20)(21,23,27,25)(22,28,31,29)(24,35,30,32)(26,33,34,36);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24,27,30)(22,26,31,34)(23,32,25,35)(28,36,29,33);(1,2,3,4)(21,25,27,23)(22,29,31,28)(24,35,30,32)(26,33,34,36);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,17,18,20)(21,26)(22,30)(23,33)(24,31)(25,36)(27,34)(28,32)(29,35)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=i,e^2=gh,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, 1204930);