#8105400

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
prime radical	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,11)(4,9)(5,10)(6,8)(7,13)(12,16)(14,15)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,31)(29,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,46)(42,47)(45,48);(1,3)(2,7)(4,14)(5,12)(6,13)(8,11)(9,16)(10,15)(17,19)(18,22)(20,29)(21,26)(23,25)(24,32)(27,30)(28,31)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,13)(12,15)(14,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,31)(28,32)(29,30)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48);(2,8)(7,11)(9,10)(15,16)(18,23)(19,26)(24,28)(27,29)(34,40)(35,43)(39,44)(42,45);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,15,11,16)(17,20,21,30)(18,24,23,28)(19,27,26,29)(22,31,25,32)(34,40)(38,41)(39,44)(47,48);(1,5,6,4)(2,10,8,9)(3,12,13,14)(7,15,11,16)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=gi,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, 8012596);