#832778

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1638 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)(3,4)(5,6)(7,8)(10,11)(14,15)(18,21)(22,24)(25,26,28,30)(27,29,31,32);(2,6)(4,7)(25,26,28,30)(27,29,31,32);(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,3,5,8)(2,7,6,4)(9,11,13,10)(12,15,16,14);(1,4,5,7)(2,3,6,8)(9,12,13,16)(10,14,11,15)(17,19,20,23)(18,22,21,24)(25,27,28,31)(26,29,30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fi,d^2=i,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, 739974);