#8127493

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	#45576	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034689);