#1097461

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

2124 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)(7,8)(9,11)(21,22)(23,30)(24,28)(25,33)(26,27)(29,34)(31,32)(35,36);(5,6,10,12)(7,11,8,9)(13,14,17,16)(15,19,20,18)(22,27)(23,28)(25,34)(32,36);(5,7,10,8)(6,11,12,9)(13,15,17,20)(14,18,16,19)(21,23)(22,24)(25,31)(26,28)(27,30)(29,32)(33,36)(34,35);(1,2)(3,4)(5,8,10,7)(6,9,12,11)(21,24)(22,28)(23,27)(25,32)(26,30)(29,35)(31,33)(34,36);(5,9)(6,7)(8,12)(10,11)(13,16,17,14)(15,19,20,18)(21,25)(22,29)(23,31)(24,32)(26,34)(27,33)(28,35)(30,36)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1004657);