#2840059

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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2632 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,c^2=f,d^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, 2747255);