#4029232

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=gi,d^2=fgi,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, 3936428);