#8360007

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

4091 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8267203);