#9203710

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	#55697	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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)(3,7)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18,23,25)(19,28,22,24)(20,26,31,30)(21,27,32,29)(33,34,39,44)(35,45,47,40)(36,46,48,41)(37,42,43,38);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19,23,22)(18,24,25,28)(20,29,31,27)(21,26,32,30)(33,34,39,44)(35,45,47,40)(36,41,48,46)(37,38,43,42);(3,10)(7,8)(11,13)(12,16)(18,25)(20,31)(24,28)(27,29)(33,35,39,47)(34,40,44,45)(36,43,48,37)(38,46,42,41);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,20)(18,26)(19,29)(21,28)(22,27)(23,31)(24,32)(25,30)(33,36,39,48)(34,41,44,46)(35,37,47,43)(38,40,42,45);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,21,23,32)(18,27,25,29)(19,30,22,26)(20,24,31,28)(33,37)(34,42)(35,36)(38,44)(39,43)(40,41)(45,46)(47,48);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,22,23,19)(18,28,25,24)(20,27,31,29)(21,30,32,26)(33,38,39,42)(34,43,44,37)(35,46,47,41)(36,40,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9110906);