#8746054

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	#52649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3947 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=i,e^2=gh,f^2=gh,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, 8653250);