#2830170

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1803 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=fgh,d^2=fh,e^2=g,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, 2737366);