#4957773

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

2182 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)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(19,23)(22,24)(25,26,29,30)(27,31,32,28);(1,2)(3,6)(4,5)(7,8)(17,18,21,20)(19,22,23,24)(25,27,29,32)(26,28,30,31);(2,5)(6,7)(9,10,12,14)(11,13,15,16)(17,19,21,23)(18,22,20,24)(25,26,29,30)(27,31,32,28);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,28)(29,30)(31,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,24,23,22)(25,28)(26,27)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4864969);