#8127349

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4234 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)(3,8)(4,10)(5,15)(6,16)(7,9)(11,14)(12,13)(17,18,23,25)(19,28,31,24)(20,32,27,22)(21,30,29,26)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,3)(2,8)(4,7)(5,11)(6,12)(9,10)(13,16)(14,15)(17,19,23,31)(18,24,25,28)(20,21,27,29)(22,30,32,26)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(2,9)(3,4)(5,16)(12,14)(18,25)(20,27)(21,29)(24,28)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(1,4,7,3)(2,10,9,8)(5,13,16,11)(6,14,15,12)(17,20,23,27)(18,22,25,32)(19,29,31,21)(24,26,28,30)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,5)(2,6)(3,11)(4,13)(7,16)(8,12)(9,15)(10,14)(17,21,23,29)(18,26,25,30)(19,27,31,20)(22,28,32,24)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47);(1,6)(2,5)(3,12)(4,14)(7,15)(8,11)(9,16)(10,13)(17,22)(18,27)(19,30)(20,25)(21,28)(23,32)(24,29)(26,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=h,e^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, 8034545);