#792152

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fghi,c^2=fgh,d^2=f,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, 699348);