#618475

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

2206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,d^2=h,e^2=f,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, 525671);