#2329246

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=gh,d^2=hi,e^2=h,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, 2236442);