#8712697

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	#41122	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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)(4,6)(5,14)(7,9)(10,12)(11,15)(13,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34,39,44)(35,45,47,40)(36,38,43,41)(37,48,46,42);(1,3,7,11)(2,8,9,15)(4,5,13,12)(6,14,16,10)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,35)(34,40)(36,37)(38,42)(39,47)(41,48)(43,46)(44,45);(2,9)(3,11)(4,13)(10,14)(19,28)(20,29)(24,30)(25,32)(33,36,39,43)(34,41,44,38)(35,37,47,46)(40,48,45,42);(1,4,7,13)(2,6,9,16)(3,12,11,5)(8,10,15,14)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,36)(34,41)(35,46)(37,47)(38,44)(39,43)(40,42)(45,48);(1,5,7,12)(2,10,9,14)(3,13,11,4)(6,15,16,8)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,37,39,46)(34,42,44,48)(35,36,47,43)(38,45,41,40);(1,6)(2,4)(3,14)(5,8)(7,16)(9,13)(10,11)(12,15)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,38,39,41)(34,43,44,36)(35,42,47,48)(37,40,46,45)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8619893);