#10171375

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

4090 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,24)(20,22)(21,26)(23,25)(27,29)(28,31)(30,32)(33,34,39,44)(35,43,38,40)(36,37,42,41)(45,46,47,48);(3,9)(7,13)(10,15)(14,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,31,32,29)(33,35,39,38)(34,40,44,43)(36,45,42,47)(37,48,41,46);(2,6)(7,13)(8,11)(14,16)(18,25)(19,20)(26,32)(27,28)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,46,43,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,22,25,24)(21,27,30,28)(26,29,32,31)(33,36)(34,41)(35,45)(37,44)(38,47)(39,42)(40,46)(43,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,31,24,29)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,45,43,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,28)(27,32)(30,31)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,46,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=i,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, 10078571);