#647644

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1974 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)(5,6,9,12)(7,11,8,10)(15,19)(18,20)(21,22,25,24)(23,26,28,27);(5,7,9,8)(6,10,12,11)(13,14,17,16)(15,18,19,20)(21,23,25,28)(22,26,24,27);(1,2,3,4)(7,8)(10,11)(21,23)(22,26)(24,27)(25,28);(5,8,9,7)(6,11,12,10)(13,15)(14,18)(16,20)(17,19)(21,22,25,24)(23,26,28,27);(5,6)(7,10)(8,11)(9,12)(13,16,17,14)(15,20,19,18)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=g,d^2=h,e^2=hi,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, 554840);