#10480804

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	#57967	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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)(3,4)(5,9)(6,10)(7,8)(11,12)(13,16)(14,15)(20,28)(23,29)(26,32)(30,31)(35,42)(39,46)(43,47)(44,48);(2,8)(4,9)(10,15)(12,16)(17,18)(19,24)(20,29)(21,25)(22,27)(23,28)(26,30)(31,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3,7,5)(2,4,8,9)(6,11,14,13)(10,12,15,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,32)(29,31)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,4,7,9)(2,3,8,5)(6,12,14,16)(10,11,15,13)(17,20,21,28)(18,23,25,29)(19,26,27,32)(22,30,24,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5,7,3)(2,9,8,4)(6,11,14,13)(10,12,15,16)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,32)(29,30)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gi,e^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, 10388000);