#2765055

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
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

2711 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2672251);