#4951402

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1997 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)(10,13)(11,15)(17,18)(19,23)(20,22)(21,24)(27,29)(31,32);(9,10,12,13)(11,14,15,16)(17,19,22,21)(18,23,20,24)(25,26,30,28)(27,32,29,31);(2,6)(3,8)(11,15)(14,16)(18,20)(23,24)(25,27,30,29)(26,31,28,32);(1,3,5,8)(2,7,6,4)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,24,21,23)(25,28,30,26)(27,31,29,32);(1,4,5,7)(2,8,6,3)(9,10)(11,14)(12,13)(15,16)(17,21,22,19)(18,24,20,23)(25,29,30,27)(26,31,28,32)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=fg,e^2=g,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, 4858598);