#1100291

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2780 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=fg,d^2=g,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, 1007487);