#2084436

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fg,d^2=h,e^2=f,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, 1991632);