#2806192

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

2288 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)(9,10,14,15)(11,13,12,16)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32);(6,8)(10,15)(13,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31);(5,6)(7,8)(9,11,14,12)(10,13,15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32);(1,2)(3,4)(9,12,14,11)(10,13,15,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32);(9,13)(10,12)(11,15)(14,16)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=f,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, 2713388);