#9070544

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

4092 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8977740);