#2738369

order	512
nilpotency class	2
isoclinism	isoclinic class of #2735949
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2298 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)(14,18)(16,19)(21,22)(23,29)(24,28)(25,26)(27,30)(31,36)(32,35)(33,34);(7,11)(10,12)(14,18)(15,20)(22,26)(23,30)(28,34)(31,32);(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,20,16)(21,22)(23,29)(24,28)(25,26)(27,30)(31,36)(32,35)(33,34);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,18)(17,20)(21,23)(22,27)(24,32)(25,30)(26,29)(28,36)(31,33)(34,35);(5,6)(7,10)(8,9)(11,12)(13,16,17,19)(14,15,18,20)(21,24,25,33)(22,28,26,34)(23,31,30,32)(27,35,29,36)

Polycyclic presentation:

2^9|c^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

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, 2645565);