#8926355

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3784 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=g,e^2=h,f^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, 8833551);