#8813997

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

4182 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)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,26)(20,23)(21,24)(22,28)(25,31)(27,29)(30,32)(33,34)(35,37)(36,40)(38,42)(39,44)(41,46)(43,47)(45,48);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,15,16,13)(17,19,23,24)(18,21,20,26)(22,30,31,27)(25,32,28,29)(33,35,38,44)(34,39,42,37)(36,43,46,48)(40,45,41,47);(2,9)(3,4)(5,11)(13,15)(18,20)(21,26)(25,28)(29,32)(33,36,38,46)(34,40,42,41)(35,43,44,48)(37,47,39,45);(1,4,7,3)(2,10,9,8)(5,14,11,12)(6,15,16,13)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,32,30,29)(33,36)(34,41)(35,43)(37,45)(38,46)(39,47)(40,42)(44,48);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,13,10,15)(17,21,23,26)(18,24,20,19)(22,29,31,32)(25,30,28,27)(33,37)(34,35)(36,45)(38,39)(40,48)(41,43)(42,44)(46,47);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,14)(10,12)(17,22,23,31)(18,25,20,28)(19,27,24,30)(21,29,26,32)(33,35)(34,37)(36,43)(38,44)(39,42)(40,47)(41,45)(46,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8721193);