#2844113

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1974 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,24)(21,23)(25,26,29,31)(27,30,32,28);(1,2,5,7)(3,8,4,6)(11,15)(13,16)(17,19,21,24)(18,22,23,20)(25,27,29,32)(26,28,31,30);(1,3,5,4)(2,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,28,29,30)(26,27,31,32);(1,4,5,3)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,28)(29,31)(30,32);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=f,d^2=h,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, 2751309);