#8930818

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	#55057	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4186 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)(3,8)(4,12)(5,7)(6,11)(9,13)(10,14)(15,16)(19,25)(23,30)(26,28)(31,32)(33,34)(35,39)(36,37)(38,42)(40,45)(41,48)(43,44)(46,47);(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)(34,39)(36,43)(37,44)(38,45)(40,42)(41,46)(47,48);(2,7)(4,13)(6,8)(10,16)(18,22)(19,25)(24,27)(26,28)(33,35,38,45)(34,40,42,39)(36,43,41,46)(37,47,48,44);(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)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(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)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(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)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)

Polycyclic presentation:

2^9|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, 8838014);