#617351

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2706 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)(5,6,10,11)(7,9,8,12)(15,19)(18,20)(21,22,24,26)(23,25,27,28);(6,11)(7,8)(13,14,17,16)(15,18,19,20)(21,22,24,26)(23,25,27,28);(5,7,10,8)(6,12,11,9)(13,15)(14,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,8,10,7)(6,9,11,12)(13,15)(14,18)(16,20)(17,19);(5,9)(6,7)(8,11)(10,12)(13,16,17,14)(15,20,19,18)(21,23,24,27)(22,25,26,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,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, 524547);