#9206604

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	#40276	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,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, 9113800);