#9980357

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	#49658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9887553);