#1031577

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 938773);