#2832377

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2168 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,d^2=f,e^2=gh,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, 2739573);