#4456926

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 4194304
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

2382 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4364122);