#8417127

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3899 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=ghi,e^2=gi,f^2=h,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, 8324323);