#2806523

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2532 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)(9,10)(11,15)(12,16)(13,14)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,31,30,32);(6,8)(10,14)(11,16)(18,22)(19,26)(20,29)(31,32);(5,6)(7,8)(9,11,13,16)(10,15,14,12)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32);(1,2)(3,4)(9,12)(10,11)(13,15)(14,16)(17,19,21,26)(18,23,22,25)(20,30,29,27)(24,32,28,31);(9,10)(11,15)(12,16)(13,14)(17,20,21,29)(18,24,22,28)(19,27,26,30)(23,31,25,32)

Polycyclic presentation:

2^9|a^2=g,c^2=fg,d^2=g,e^2=g,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, 2713719);