#4331829

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2278 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,20,23,22)(25,26)(27,31)(28,32)(29,30);(1,2,5,6)(3,4,8,7)(17,19,21,23)(18,22,24,20)(25,27,29,32)(26,28,30,31);(2,6)(3,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,22)(18,23,24,19)(25,26,29,30)(27,31,32,28);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32);(1,4)(2,3)(5,7)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4239025);