#1168414

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

2580 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=fg,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, 1075610);