#8324252

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

4071 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)(3,5)(4,12)(6,8)(7,11)(9,15)(10,16)(13,14)(17,18)(19,20)(21,29)(22,23)(24,31)(25,28)(26,30)(27,32)(33,34,39,42)(35,38,41,47)(36,37,40,46)(43,44,45,48);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,19,22,25)(18,20,23,28)(21,26,31,32)(24,27,29,30)(33,35)(34,38)(36,43)(37,44)(39,41)(40,45)(42,47)(46,48);(2,8)(5,7)(9,12)(14,16)(18,23)(19,25)(24,29)(26,32)(35,41)(38,47)(43,45)(44,48);(1,4,6,15)(2,9,8,12)(3,10,11,13)(5,14,7,16)(17,20)(18,19)(21,30)(22,28)(23,25)(24,32)(26,29)(27,31)(33,36)(34,37)(35,43)(38,44)(39,40)(41,45)(42,46)(47,48);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,21)(18,24)(19,26)(20,27)(22,31)(23,29)(25,32)(28,30)(33,37)(34,40)(35,44)(36,42)(38,45)(39,46)(41,48)(43,47);(1,5,6,7)(2,3,8,11)(4,14,15,16)(9,10,12,13)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,38,39,47)(34,41,42,35)(36,44,40,48)(37,45,46,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,f^2=hi,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, 8231448);