#2128056

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

1900 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=h,d^2=fg,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, 2035252);