#8773035

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	#41307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8680231);