#9490196

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

4362 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:

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

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,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9397392);