#9206601

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3816 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gh,e^2=ghi,f^2=gh,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, 9113797);