#814022

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
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

1971 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)(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18,22,21)(19,24,23,20)(25,26,29,28)(27,31,32,30);(2,6)(4,7)(10,15)(11,13)(17,18,22,21)(19,20,23,24)(25,26,29,28)(27,30,32,31);(9,11,14,13)(10,16,15,12)(17,19)(18,20)(21,24)(22,23)(25,27,29,32)(26,30,28,31);(1,3,5,8)(2,7,6,4)(9,12)(10,11)(13,15)(14,16)(17,20,22,24)(18,23,21,19)(25,26,29,28)(27,30,32,31);(1,4,5,7)(2,3,6,8)(9,13,14,11)(10,16,15,12)(17,21,22,18)(19,24,23,20)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=fghi,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 721218);