#8691803

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8598999);