#8104716

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8011912);