#10153081

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,23,25)(19,28,22,24)(20,29,30,26)(21,27,32,31)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,23,22)(18,24,25,28)(20,27,30,31)(21,26,32,29)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(2,4)(3,10)(9,12)(11,16)(18,25)(20,30)(24,28)(27,31)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20,23,30)(18,26,25,29)(19,27,22,31)(21,24,32,28)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21,23,32)(18,27,25,31)(19,29,22,26)(20,28,30,24)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,22,23,19)(18,28,25,24)(20,31,30,27)(21,29,32,26)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=g,e^2=gi,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, 10060277);