#10181234

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	#56351	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10088430);