#8712765

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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)(3,5)(4,12)(6,9)(7,11)(8,13)(10,16)(14,15)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34)(35,41)(36,43)(37,39)(38,42)(40,44)(45,48)(46,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,16)(12,14)(13,15)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,35,40,46)(34,41,44,47)(36,37,38,45)(39,42,48,43);(4,13)(8,12)(10,15)(14,16)(19,28)(20,29)(24,30)(25,32)(33,36,40,38)(34,42,44,43)(35,45,46,37)(39,47,48,41);(1,4)(2,8)(3,10)(5,14)(6,13)(7,16)(9,12)(11,15)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,37,40,45)(34,39,44,48)(35,36,46,38)(41,43,47,42);(1,5,6,7)(2,3,9,11)(4,14,13,16)(8,10,12,15)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,38,40,36)(34,43,44,42)(35,45,46,37)(39,47,48,41);(1,3)(2,5)(4,15)(6,11)(7,9)(8,16)(10,13)(12,14)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,39)(34,37)(35,42)(36,47)(38,41)(40,48)(43,46)(44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=h,e^2=gh,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, 8619961);