#2816141

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2437 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=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, 2723337);