#987527

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fghi,c^2=i,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 894723);