#4970147

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

1584 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=fi,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, 4877343);