#4379763

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2870 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)(5,6,9,7)(8,11,12,10)(15,19)(18,20)(21,22)(23,26)(24,28)(25,27);(6,7)(10,11)(14,17)(18,20)(21,23,25,28)(22,26,27,24);(5,6,9,7)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,7,9,6)(8,11,12,10);(5,8,9,12)(6,10,7,11)(13,15)(14,18)(16,19)(17,20)(21,24)(22,23)(25,26)(27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

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, 4286959);