#741877

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1932 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gi,c^2=fgi,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 649073);