#8334564

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	#41495	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4001 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8241760);