#10504595

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

4011 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,e^2=gi,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, 10411791);