#8253888

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
prime radical	#55173	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,15)(13,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3,5,11)(2,6,7,9)(4,13,14,10)(8,16,12,15)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(2,7)(6,9)(8,12)(15,16)(18,22)(23,25)(24,28)(30,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,15)(12,16)(13,14)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4,5,14)(2,8,7,12)(3,10,11,13)(6,15,9,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)

Polycyclic presentation:

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

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, 8161084);