#8750002

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#52977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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)(3,8)(4,7)(5,13)(6,10)(9,15)(11,12)(14,16)(17,18,23,27)(19,24,22,26)(20,25,31,21)(28,30,29,32)(33,34,39,40)(35,43,37,41)(36,42,46,38)(44,48,47,45);(1,3,7,11)(2,8,4,12)(5,10,15,14)(6,9,16,13)(17,19,23,22)(18,24,27,26)(20,29,31,28)(21,30,25,32)(34,40)(36,46)(41,43)(44,47);(2,4)(3,11)(9,13)(10,14)(17,20,23,31)(18,25,27,21)(19,28,22,29)(24,30,26,32)(33,35,39,37)(34,41,40,43)(36,47,46,44)(38,45,42,48);(1,4,7,2)(3,8,11,12)(5,13,15,9)(6,10,16,14)(17,20)(18,21)(19,29)(22,28)(23,31)(24,30)(25,27)(26,32)(33,36,39,46)(34,42,40,38)(35,44,37,47)(41,45,43,48);(1,5)(2,9)(3,10)(4,13)(6,12)(7,15)(8,16)(11,14)(17,21)(18,20)(19,30)(22,32)(23,25)(24,29)(26,28)(27,31)(33,37,39,35)(34,43,40,41)(36,47,46,44)(38,45,42,48);(1,6,7,16)(2,10,4,14)(3,9,11,13)(5,8,15,12)(17,22,23,19)(18,26,27,24)(20,29,31,28)(21,30,25,32)(33,38,39,42)(34,36,40,46)(35,45,37,48)(41,47,43,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657198);