#4867263

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2707 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)(6,10)(7,11)(15,19)(17,20)(22,25)(23,27);(5,6,9,10)(7,8,11,12)(13,14,16,18)(15,17,19,20)(23,27)(26,28);(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(21,23,24,27)(22,26,25,28);(5,8)(6,11)(7,10)(9,12)(13,15)(14,17)(16,19)(18,20)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fh,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4774459);