#8831181

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	#49859	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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)(6,10)(11,15)(14,16)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(33,34,39,44)(35,42,37,40)(36,46,45,41)(38,43,47,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,10)(11,16)(14,15)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(33,35,39,37)(34,40,44,42)(36,47,45,38)(41,48,46,43);(1,2,5,8)(3,6,9,10)(4,7,13,12)(11,14,15,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,36)(34,41)(35,38)(37,47)(39,45)(40,43)(42,48)(44,46);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,13)(14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,37,39,35)(34,42,44,40)(36,47,45,38)(41,48,46,43);(1,3)(2,6)(4,11)(5,9)(7,14)(8,10)(12,16)(13,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,38,39,47)(34,43,44,48)(35,45,37,36)(40,46,42,41);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8738377);