#8156996

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

4303 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)(3,8)(4,10)(5,14)(6,13)(7,9)(11,15)(12,16)(17,18,22,24)(19,25,20,23)(21,29,30,26)(27,31,28,32)(36,44)(39,46)(42,48)(45,47);(1,3)(2,8)(4,7)(5,13)(6,14)(9,10)(11,16)(12,15)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(2,9)(3,4)(11,14)(12,13)(18,24)(23,25)(26,29)(31,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,4,7,3)(2,10,9,8)(5,13,15,12)(6,11,16,14)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,5)(2,11)(3,13)(4,12)(6,10)(7,15)(8,16)(9,14)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,6)(2,12)(3,14)(4,11)(5,10)(7,16)(8,15)(9,13)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8064192);