#721555

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,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, 628751);