#4980320

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1553 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=g,e^2=h,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, 4887516);