#9243842

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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)(3,9)(4,12)(5,6)(7,8)(10,14)(11,13)(15,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(2,8)(6,16)(9,12)(11,14)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(1,3,7,4)(2,9,8,12)(5,13,15,10)(6,11,16,14)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,4,7,3)(2,9,8,12)(5,13,15,10)(6,14,16,11)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32);(1,5)(2,6)(3,10)(4,13)(7,15)(8,16)(9,14)(11,12)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,6)(2,5)(3,11)(4,14)(7,16)(8,15)(9,13)(10,12)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9151038);