#2097112

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
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

1688 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)(7,8)(9,12)(10,11)(21,22,25,28)(23,26,30,34)(24,31,32,27)(29,35,36,33);(5,7)(6,9)(8,10)(11,12)(14,17)(18,20)(21,23,25,30)(22,26,28,34)(24,29,32,36)(27,33,31,35);(5,8,11,9)(6,12,10,7)(13,14,16,17)(15,18,19,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(1,2)(3,4)(5,9,11,8)(6,12,10,7)(21,23)(22,26)(24,29)(25,30)(27,33)(28,34)(31,35)(32,36);(5,10,11,6)(7,8,12,9)(13,15,16,19)(14,18,17,20)(21,24,25,32)(22,27,28,31)(23,29,30,36)(26,33,34,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=h,d^2=f,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, 2004308);