#8810338

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

4093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=g,e^2=g,f^2=gi,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, 8717534);