#8839502

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	#58380	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=gi,e^2=g,f^2=g,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, 8746698);