#8831161

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

4266 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)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18,22,24)(19,26,29,23)(20,30,25,21)(27,32,31,28)(35,41)(38,42)(43,47)(46,48);(2,7)(5,9)(8,10)(13,16)(17,19,22,29)(18,23,24,26)(20,27,25,31)(21,32,30,28)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(3,11)(8,10)(12,14)(13,16)(18,24)(20,25)(23,26)(27,31)(33,34,37,40)(35,38,41,42)(36,39,45,44)(43,46,47,48);(1,3,6,11)(2,8,7,10)(4,14,15,12)(5,13,9,16)(17,20,22,25)(18,21,24,30)(19,27,29,31)(23,32,26,28)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,45)(46,47);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,19)(18,23)(20,31)(21,28)(22,29)(24,26)(25,27)(30,32)(33,35)(34,38)(36,43)(37,41)(39,46)(40,42)(44,48)(45,47);(1,5,6,9)(2,4,7,15)(3,13,11,16)(8,14,10,12)(17,21)(18,25)(19,28)(20,24)(22,30)(23,27)(26,31)(29,32)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

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

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, 8738357);