#8227372

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	#56312	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3996 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=hi,e^2=gi,f^2=g,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, 8134568);