#8253892

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

4267 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)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,18,23,22)(19,25,20,24)(21,26,30,29)(27,32,28,31)(34,40)(35,37)(36,44)(43,48);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,19,23,20)(18,24,22,25)(21,28,30,27)(26,32,29,31)(33,34,39,40)(35,41,37,47)(36,45,44,38)(42,43,46,48);(2,6)(7,9)(8,12)(14,15)(17,19,23,20)(18,25,22,24)(21,27,30,28)(26,32,29,31)(33,35,39,37)(34,41,40,47)(36,42,44,46)(38,48,45,43);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,23,19)(18,25,22,24)(21,28,30,27)(26,32,29,31)(33,36)(34,38)(35,42)(37,46)(39,44)(40,45)(41,48)(43,47);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,37,39,35)(34,41,40,47)(36,46,44,42)(38,48,45,43);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,38)(34,36)(35,43)(37,48)(39,45)(40,44)(41,46)(42,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^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, 8161088);