#9910068

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	#50522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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)(3,10,4,5)(6,8,15,16)(11,14,12,13)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34)(35,40)(36,44)(37,38)(39,43)(41,47)(42,48)(45,46);(1,3,7,4)(2,5,9,10)(6,11,15,12)(8,13,16,14)(19,25)(22,29)(26,31)(30,32)(33,35)(34,40)(36,37)(38,44)(39,45)(41,48)(42,47)(43,46);(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,36,39,47)(34,41,43,44)(35,42,45,37)(38,46,48,40);(1,4,7,3)(2,5,9,10)(6,12,15,11)(8,13,16,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,41)(35,37)(38,46)(39,47)(40,48)(42,45)(43,44);(1,5)(2,3)(4,9)(6,14)(7,10)(8,12)(11,16)(13,15)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,37)(34,38)(35,36)(39,42)(40,44)(41,46)(43,48)(45,47);(1,6)(2,8)(3,11)(4,12)(5,13)(7,15)(9,16)(10,14)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,38)(34,42)(35,44)(36,46)(37,43)(39,48)(40,47)(41,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9817264);