#625791

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5950 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)(6,7)(13,14,18,15)(16,21,23,19)(17,22,25,20)(24,26,28,27);(10,12)(14,15)(16,23)(20,22)(24,28);(5,6,8,7)(9,10)(11,12)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(1,2,3,4)(5,7,8,6)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27);(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=i,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, 532987);