#8844578

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	#45468	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8751774);