#160344

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=eg,c^2=efgh,d^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

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, 67540);