#1271568

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1756 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ghi,c^2=fg,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=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, 1178764);