#617350

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

2224 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)(13,14,16,18)(15,17,19,20)(21,22)(23,27)(24,26)(25,32)(28,33)(29,30)(31,36)(34,35);(5,6,9,8)(7,10,11,12)(13,14,16,18)(15,17,19,20)(22,24)(23,29)(28,32)(31,35);(5,7)(6,10)(8,12)(9,11)(13,14)(15,17)(16,18)(19,20)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34);(5,8,9,6)(7,12,11,10)(13,15,16,19)(14,17,18,20)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=f,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, 524546);