#8533536

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	#52408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440732);