#621997

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

2130 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 529193);