#4452830

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 1048576
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

2162 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=f,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, 4360026);