#9490112

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	#54179	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4001 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(34,39)(36,40)(41,47)(44,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32)(33,34,38,39)(35,40,43,36)(37,41,46,47)(42,48,45,44);(2,3)(7,9)(8,10)(14,15)(18,23)(20,27)(21,29)(26,31)(33,35,38,43)(34,40,39,36)(37,45,46,42)(41,44,47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,31)(30,32)(33,36)(34,35)(37,44)(38,40)(39,43)(41,42)(45,47)(46,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,47)(45,46);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,37,38,46)(34,41,39,47)(35,42,43,45)(36,44,40,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=h,e^2=h,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, 9397308);