#8137101

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

4084 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=g,e^2=gh,f^2=h,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, 8044297);