#8712127

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	#41003	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,e^2=g,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, 8619323);