#9712398

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	#49770	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9619594);