#829504

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1900 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,d^2=fhi,e^2=fg,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, 736700);