#10480878

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	#51946	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) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10388074);