#8125477

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

4253 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)(3,11)(4,14)(5,13)(6,7)(8,12)(9,15)(10,16)(17,18,22,24)(19,23,28,26)(20,25,30,32)(21,29,31,27)(33,34,39,36)(35,40,42,37)(38,44,46,41)(43,45,48,47);(2,7)(3,12)(9,14)(10,13)(17,19)(18,23)(20,27)(21,25)(22,28)(24,26)(29,30)(31,32)(33,35,39,42)(34,40,36,37)(38,43,46,48)(41,47,44,45);(2,7)(4,15)(8,11)(10,13)(18,24)(19,28)(25,32)(27,29)(35,42)(37,40)(43,48)(45,47);(1,3)(2,8)(4,13)(5,9)(6,12)(7,11)(10,15)(14,16)(17,20,22,30)(18,25,24,32)(19,29,28,27)(21,23,31,26)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,11,16,8)(17,19)(18,26)(20,29)(21,25)(22,28)(23,24)(27,30)(31,32)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,5)(2,10)(3,9)(4,11)(6,16)(7,13)(8,15)(12,14)(17,21)(18,27)(19,25)(20,23)(22,31)(24,29)(26,30)(28,32)(33,38,39,46)(34,41,36,44)(35,43,42,48)(37,45,40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8032673);