#7707365

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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
prime radical	#51415	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4886 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)(3,6)(4,5)(7,8)(11,12)(14,16)(17,18)(19,23)(20,27)(21,30)(22,25)(24,29)(26,32)(28,31);(2,5)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,31,27,32);(9,11)(10,14)(12,13)(15,16)(17,20,22,29)(18,24,25,27)(19,26,21,28)(23,31,30,32);(10,15)(14,16)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,32,27,31);(9,12,13,11)(10,14,15,16)(17,21,22,19)(18,23,25,30)(20,28,29,26)(24,31,27,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7614561);