#719639

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2154 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)(6,9)(10,12)(13,14)(15,16)(17,19)(18,20)(21,22,26,25)(23,28,24,27);(13,15)(14,18)(16,19)(17,20)(21,23,26,24)(22,27,25,28);(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,18,19,16)(21,22,26,25)(23,27,24,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24,26,23)(22,27,25,28);(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,18)(19,20)(21,25,26,22)(23,27,24,28)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=f,d^2=fgi,e^2=fg,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, 626835);