#707418

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

2244 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)(10,11)(14,15)(17,18,21,20)(19,23,24,22)(27,31)(30,32);(2,6)(4,7)(9,10,13,11)(12,15,16,14)(18,20)(22,23)(26,29)(30,32);(2,6)(3,8)(10,11)(14,15)(17,19,21,24)(18,22,20,23)(25,26)(27,30)(28,29)(31,32);(1,3)(2,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,23)(21,24)(25,27,28,31)(26,30,29,32);(1,4,5,7)(2,3,6,8)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=hi,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, 614614);