#8813968

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	#49972	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,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, 8721164);