#8417208

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324404);