#805040

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

1752 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)(4,7)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,21,23,24)(27,31)(30,32);(2,6)(7,8)(9,11,14,12)(10,13,16,15)(17,18,20,22)(19,21,23,24)(25,26,29,28)(27,30,31,32);(1,3,5,4)(2,7,6,8)(9,11,14,12)(10,15,16,13)(17,18)(19,21)(20,22)(23,24)(25,27,29,31)(26,30,28,32);(1,4,5,3)(2,7,6,8)(9,12,14,11)(10,15,16,13);(1,2)(3,7)(4,8)(5,6)(9,13,14,15)(10,12,16,11)(17,19,20,23)(18,21,22,24)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=ghi,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, 712236);