#8118827

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	#37912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026023);