#4970143

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

1650 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)(3,8)(4,7)(5,6)(10,12)(11,16)(19,23)(21,24)(25,26,29,31)(27,30,28,32);(2,5)(7,8)(9,10,14,12)(11,13,16,15)(25,27,29,28)(26,30,31,32);(1,3,6,4)(2,7,5,8)(9,11,14,16)(10,15,12,13)(17,18,20,22)(19,21,23,24)(25,28,29,27)(26,30,31,32);(1,4,6,3)(2,8,5,7)(9,12,14,10)(11,13,16,15)(17,19)(18,21)(20,23)(22,24)(25,26)(27,30)(28,32)(29,31);(1,5,6,2)(3,8,4,7)(9,13)(10,11)(12,16)(14,15)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=i,d^2=f,e^2=fh,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, 4877339);