#2327630

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

1702 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)(5,6,11,12)(7,8,10,9)(13,14,17,16)(15,18,19,20)(21,22)(23,25)(24,31)(26,27)(28,32)(29,34)(30,36)(33,35);(5,7,11,10)(6,9,12,8)(15,19)(18,20)(22,27)(23,29)(24,32)(33,36);(5,8,11,9)(6,7,12,10)(13,15,17,19)(14,18,16,20)(21,23,26,29)(22,25,27,34)(24,30,32,35)(28,33,31,36);(1,2,3,4)(5,9,11,8)(6,7,12,10)(13,14,17,16)(15,18,19,20)(21,24)(22,28)(23,30)(25,33)(26,32)(27,31)(29,35)(34,36);(5,10,11,7)(6,8,12,9)(13,16,17,14)(15,20,19,18)(21,25)(22,23)(24,33)(26,34)(27,29)(28,30)(31,35)(32,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=h,d^2=hi,e^2=fgh,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, 2234826);