#636914

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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

2246 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)(7,11)(10,12)(13,14)(15,17)(16,19)(18,20)(21,22,26,27)(23,24,28,32)(25,33,34,29)(30,36,35,31);(5,6,9,8)(7,10,11,12)(13,15,18,19)(14,17,20,16)(22,27)(23,28)(29,33)(30,35);(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,36)(34,35);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16,18,17)(14,19,20,15)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,33)(34,36);(5,8,9,6)(7,12,11,10)(13,17,18,16)(14,15,20,19)(21,25,26,34)(22,29,27,33)(23,30,28,35)(24,31,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 544110);