#8926526

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	#46930	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833722);