#4928776

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
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	#14762	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2794 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,28)(24,26)(25,27);(6,9)(10,12)(13,14,16,17)(15,18,20,19)(21,23,25,24)(22,26,27,28);(14,17)(18,19)(22,27)(26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,28)(24,26)(25,27);(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,18,17,19)(21,24,25,23)(22,26,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4835972);