#626055

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

1930 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)(6,10)(7,12)(13,14,16,18)(15,17,19,20)(21,22,26,27)(23,28,29,24)(25,33,34,30)(31,32,36,35);(13,14,16,18)(15,17,19,20)(22,27)(24,28)(30,33)(32,35);(5,6)(7,11)(8,12)(9,10)(13,14)(15,17)(16,18)(19,20)(21,23,26,29)(22,28,27,24)(25,31,34,36)(30,35,33,32);(1,2,3,4)(5,7,9,12)(6,11,10,8)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31);(5,8,9,11)(6,12,10,7)(13,15,16,19)(14,17,18,20)(21,25,26,34)(22,30,27,33)(23,31,29,36)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=f,d^2=i,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, 533251);