#8137635

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

4237 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)(3,10)(4,9)(5,14)(6,8)(7,11)(12,16)(13,15)(18,23)(19,28)(20,30)(21,31)(35,42)(37,45)(39,40)(41,44);(1,3,7,6)(2,8,11,10)(4,12,5,13)(9,15,14,16)(17,18,22,23)(19,24,28,25)(20,26,30,32)(21,27,31,29)(33,34)(35,40)(36,38)(37,41)(39,42)(43,47)(44,45)(46,48);(1,4,7,5)(2,9,11,14)(3,12,6,13)(8,15,10,16)(17,19)(18,24)(20,29)(21,26)(22,28)(23,25)(27,30)(31,32)(33,35)(34,39)(36,40)(37,43)(38,42)(41,46)(44,47)(45,48);(1,5,7,4)(2,9,11,14)(3,13,6,12)(8,15,10,16)(17,19)(18,25)(20,27)(21,26)(22,28)(23,24)(29,30)(31,32)(33,36,38,34)(35,40,42,39)(37,44,45,41)(43,47,48,46);(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,20)(18,26)(19,29)(21,24)(22,30)(23,32)(25,31)(27,28)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,6,7,3)(2,10,11,8)(4,13,5,12)(9,16,14,15)(17,21,22,31)(18,27,23,29)(19,26,28,32)(20,24,30,25)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=gh,f^2=g,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, 8044831);