#5609955

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
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

2050 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)(7,9,11,12)(13,14,17,19)(15,16,20,18)(21,22)(23,29)(24,28)(25,26)(27,31)(30,36)(32,35)(33,34);(5,7,8,11)(6,9,10,12)(13,14,17,19)(15,16,20,18)(22,26)(27,29)(28,34)(35,36);(6,10)(9,12)(13,15,17,20)(14,18,19,16)(21,22)(23,29)(24,28)(25,26)(27,31)(30,36)(32,35)(33,34);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23,25,31)(22,27,26,29)(24,32,33,30)(28,36,34,35);(5,7)(6,9)(8,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,24,25,33)(22,28,26,34)(23,30,31,32)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5517151);