#617352

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	#15273	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2909 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)(7,9)(8,11)(10,12)(15,19)(18,20)(21,22,24,26)(23,25,27,28);(6,7)(10,11)(13,14,17,16)(15,18,19,20)(21,22,24,26)(23,25,27,28);(5,6,9,7)(8,10,12,11)(13,15)(14,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,20)(17,19);(5,8)(6,10)(7,11)(9,12)(13,16,17,14)(15,20,19,18)(21,23,24,27)(22,25,26,28)

Polycyclic presentation:

2^9|a^2=fh,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, 524548);