#3750222

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1582 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)(6,8)(10,11)(12,16)(17,18,21,22)(19,23,24,20);(9,10,13,11)(12,14,16,15)(19,24)(20,23)(25,26,29,28)(27,31,32,30);(9,10,13,11)(12,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,27,29,32)(26,30,28,31);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,14,16,15)(17,20)(18,19)(21,23)(22,24);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,20)(21,22)(23,24)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=gh,d^2=i,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3657418);