#707422

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

2192 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fhi,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

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, 614618);