#2760708

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(11,12)(14,15)(17,18)(19,20)(21,26)(22,23)(24,25)(27,28)(29,32)(30,31);(6,8)(9,10,13,16)(11,14,12,15)(18,23)(19,25)(26,31)(27,32);(5,6)(7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,23)(21,27)(22,25)(26,29)(28,31)(30,32);(1,2)(3,4)(9,10)(11,14)(12,15)(13,16)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32);(9,12,13,11)(10,15,16,14)(17,21,22,30)(18,26,23,31)(19,27,25,32)(20,28,24,29)

Polycyclic presentation:

2^9|b^2=fg,c^2=fg,d^2=f,e^2=g,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, 2667904);