#8810104

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

4196 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)(17,18,23,25)(19,29,31,24)(20,32,30,26)(21,27,28,22)(33,34,38,43)(35,42,44,39)(36,45,46,40)(37,41,47,48);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,15,16,13)(17,19,23,31)(18,24,25,29)(20,28,30,21)(22,32,27,26)(33,35,38,44)(34,39,43,42)(36,41,46,48)(37,40,47,45);(2,9)(3,4)(5,11)(13,15)(18,25)(20,30)(21,28)(24,29)(36,46)(37,47)(40,45)(41,48);(1,4,7,3)(2,10,9,8)(5,14,11,12)(6,15,16,13)(17,20,23,30)(18,26,25,32)(19,28,31,21)(22,29,27,24)(33,36)(34,40)(35,41)(37,42)(38,46)(39,47)(43,45)(44,48);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,13,10,15)(17,21,23,28)(18,27,25,22)(19,30,31,20)(24,32,29,26)(33,37)(34,41)(35,40)(36,42)(38,47)(39,46)(43,48)(44,45);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,14)(10,12)(17,22,23,27)(18,28,25,21)(19,26,31,32)(20,29,30,24)(33,35)(34,42)(36,41)(37,40)(38,44)(39,43)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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, 8717300);