#688281

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, cosocle	C₂⁴	C₂⁵
prime radical	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=fh,d^2=g,e^2=g,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, 595477);