#826420

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

1698 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fgh,c^2=fgh,d^2=fi,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, 733616);