#3067294

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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

1788 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=ghi,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

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, 2974490);