#10090758

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	#44178	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,14)(13,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,38,41)(35,37,43,39)(36,40,46,48)(42,47,44,45);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(19,25)(22,29)(26,31)(30,32)(33,35)(34,37)(36,44)(38,43)(39,41)(40,45)(42,46)(47,48);(2,4)(8,10)(9,12)(15,16)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35,38,43)(34,39,41,37)(36,42,46,44)(40,45,48,47);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35)(34,37)(36,44)(38,43)(39,41)(40,45)(42,46)(47,48);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,36)(34,40)(35,42)(37,47)(38,46)(39,45)(41,48)(43,44);(1,6,7,3)(2,10,4,8)(5,11,14,13)(9,15,12,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,37)(34,35)(36,45)(38,39)(40,44)(41,43)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=g,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, 9997954);