#857785

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

1740 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 764981);