#2826899

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

2222 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=f,d^2=gh,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, 2734095);