#1506683

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1888 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)(3,7)(4,8)(5,6)(10,12)(11,16)(17,18,22,24)(19,20,23,21)(25,26)(27,31)(28,29)(30,32);(2,6)(4,7)(9,10)(11,15)(12,13)(14,16)(17,19,22,23)(18,20,24,21)(26,29)(30,31);(9,11)(10,14)(12,15)(13,16)(17,20,22,21)(18,19,24,23)(25,26)(27,30)(28,29)(31,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,21,22,20)(18,23,24,19)(25,27,28,32)(26,30,29,31);(1,4,5,7)(2,8,6,3)(9,12,13,10)(11,15,16,14)(17,18)(19,20)(21,23)(22,24)(25,26)(27,31)(28,29)(30,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=gh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1413879);