#8535299

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

4249 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442495);