#8184876

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092072);