#696689

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2276 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=hi,d^2=g,e^2=fg,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, 603885);