#8118788

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	#52410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18)(19,20)(21,29)(22,23)(24,26)(25,30)(27,31)(28,32)(37,45)(40,47)(43,44)(46,48);(2,7)(3,9)(10,13)(11,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(2,7)(4,8)(10,13)(12,15)(18,23)(19,26)(25,29)(27,32)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,46,47,48);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,15,13,12)(17,20,22,24)(18,19,23,26)(21,28,30,31)(25,27,29,32)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,47,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,29)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|c^2=ghi,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8025984);