#2549017

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1983 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2456213);