#1169855

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=fgi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1077051);