#8271932

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4260 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(33,34,40,44)(35,45,47,41)(36,37,38,42)(39,43,46,48);(3,4)(5,14)(7,10)(8,13)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(33,35)(34,41)(36,43)(37,39)(38,48)(40,47)(42,46)(44,45);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(33,36,40,38)(34,42,44,37)(35,43,47,48)(39,41,46,45);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(33,37)(34,36)(35,46)(38,44)(39,47)(40,42)(41,48)(43,45);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32)(33,38,40,36)(34,37,44,42)(35,43,47,48)(39,41,46,45);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)(33,39,40,46)(34,43,44,48)(35,37,47,42)(36,45,38,41)

Polycyclic presentation:

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

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, 8179128);