#8930936

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8838132);