#1680016

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2304 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)(13,14,17,18)(15,19,20,16)(21,22,26,30)(23,28,24,27)(25,33,34,29)(31,35,32,36);(7,11)(10,12)(15,20)(16,19)(21,23,26,24)(22,27,30,28)(25,32,34,31)(29,36,33,35);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24,26,23)(22,28,30,27)(25,32,34,31)(29,36,33,35);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,28)(24,27)(25,29)(26,30)(31,36)(32,35)(33,34);(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,18)(19,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=f,d^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1587212);