#8285429

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

4147 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8192625);