#3067299

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,c^2=ghi,d^2=f,e^2=ghi,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, 2974495);