#758212

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2342 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=fi,e^2=fi,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, 665408);