#2125162

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1888 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=g,c^2=fg,d^2=fg,e^2=fh,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, 2032358);