#9685552

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	#53165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dg,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, 9592748);