#8933111

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	#47891	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840307);