#840673

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

1626 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)(6,8)(10,11)(14,15)(17,18,22,24)(19,21,20,23)(25,26)(27,28)(29,30)(31,32);(17,19,22,20)(18,23,24,21)(26,30)(27,32);(9,10,13,11)(12,14,16,15)(17,19,22,20)(18,21,24,23)(25,27)(26,31)(28,30)(29,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,19)(18,21,24,23)(25,28,29,31)(26,27,30,32);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,21)(18,20)(19,24)(22,23)(25,27)(26,28)(29,32)(30,31)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=fg,d^2=fgh,e^2=fhi,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, 747869);