#4975878

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3075 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,21)(19,24,20,23)(25,26,29,28)(27,31,32,30);(2,4)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,20)(18,23,21,24);(3,7)(6,8)(10,13)(14,15)(19,20)(23,24)(25,27,29,32)(26,30,28,31);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,23,21,24)(25,27)(26,30)(28,31)(29,32);(1,4,5,2)(3,8,7,6)(9,11,12,16)(10,14,13,15)(17,21,22,18)(19,23,20,24)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,d^2=f,e^2=fg,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, 4883074);