#8813970

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	#49624	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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)(3,8)(4,5)(6,7)(9,12)(10,16)(11,13)(14,15)(17,18,23,28)(19,26,21,24)(20,30,29,25)(22,27,31,32)(33,34,40,44)(35,37,38,41)(36,43,39,42)(45,46,48,47);(2,7)(5,15)(8,9)(11,16)(17,19,23,21)(18,24,28,26)(20,31,29,22)(25,32,30,27)(33,35)(34,41)(36,45)(37,44)(38,40)(39,48)(42,46)(43,47);(1,3,6,12)(2,8,7,9)(4,13,14,10)(5,11,15,16)(17,20)(18,25)(19,22)(21,31)(23,29)(24,27)(26,32)(28,30)(33,36,40,39)(34,42,44,43)(35,45,38,48)(37,47,41,46);(1,3)(2,9)(4,10)(5,11)(6,12)(7,8)(13,14)(15,16)(17,21,23,19)(18,26,28,24)(20,31,29,22)(25,32,30,27)(33,37,40,41)(34,35,44,38)(36,46,39,47)(42,48,43,45);(1,4)(2,5)(3,10)(6,14)(7,15)(8,16)(9,11)(12,13)(17,22,23,31)(18,27,28,32)(19,29,21,20)(24,30,26,25)(33,38,40,35)(34,41,44,37)(36,45,39,48)(42,47,43,46);(1,5)(2,4)(3,11)(6,15)(7,14)(8,13)(9,10)(12,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,30)(31,32)(33,39,40,36)(34,43,44,42)(35,45,38,48)(37,47,41,46)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8721166);