#8908059

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

4241 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)(4,9)(5,14)(6,7)(8,11)(12,16)(13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,34,39,41)(35,42,37,40)(36,45,46,38)(43,44,47,48);(2,7)(4,14)(8,10)(12,15)(17,19,23,20)(18,24,22,25)(21,28,30,27)(26,32,29,31)(33,35,39,37)(34,40,41,42)(36,43,46,47)(38,44,45,48);(2,7)(3,11)(5,9)(12,15)(18,22)(19,20)(26,29)(27,28)(36,46)(38,45)(43,47)(44,48);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,20,23,19)(18,25,22,24)(21,27,30,28)(26,31,29,32)(33,36)(34,38)(35,43)(37,47)(39,46)(40,44)(41,45)(42,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,21,23,30)(18,26,22,29)(19,27,20,28)(24,31,25,32)(33,37,39,35)(34,40,41,42)(36,43,46,47)(38,48,45,44);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,38)(34,36)(35,44)(37,48)(39,45)(40,43)(41,46)(42,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815255);