#2309523

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

1878 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)(7,9)(8,12)(10,11)(13,14,18,16)(15,19,17,20)(21,22,26,27)(23,28,31,35)(24,25,30,29)(32,34,36,33);(5,7)(6,8)(9,11)(10,12)(13,15,18,17)(14,19,16,20)(22,27)(23,31)(24,30)(33,34);(5,8,10,9)(6,7,11,12)(13,16,18,14)(15,19,17,20)(21,23)(22,28)(24,32)(25,34)(26,31)(27,35)(29,33)(30,36);(1,2,3,4)(5,9,10,8)(6,7,11,12)(13,17,18,15)(14,19,16,20)(21,24,26,30)(22,29,27,25)(23,32,31,36)(28,33,35,34);(5,7)(6,8)(9,11)(10,12)(21,25,26,29)(22,30,27,24)(23,33,31,34)(28,32,35,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=fgh,d^2=gi,e^2=fg,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, 2216719);