#625747

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

2086 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fghi,c^2=fgi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 532943);