#8127609

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3934 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=h,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, 8034805);