#8712423

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	#55104	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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)(3,10)(4,13)(5,15)(6,8)(7,9)(11,16)(12,14)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,44)(47,48);(1,3,7,6)(2,8,9,10)(4,14,15,11)(5,12,13,16)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(35,39)(36,42)(43,47)(44,48);(2,9)(4,15)(8,10)(11,14)(19,28)(20,29)(24,30)(25,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,4)(2,5)(3,11)(6,14)(7,15)(8,16)(9,13)(10,12)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,5)(2,4)(3,12)(6,16)(7,13)(8,14)(9,15)(10,11)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48);(1,6,7,3)(2,8,9,10)(4,11,15,14)(5,12,13,16)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,34)(35,36)(37,45)(38,41)(39,42)(40,46)(43,48)(44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gh,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, 8619619);