#9913600

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

4180 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820796);