#4030101

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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

1720 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=hi,c^2=gi,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3937297);