#8137127

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	#48458	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=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, 8044323);