#9850598

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

4108 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,22,25)(19,23,27,20)(21,29,31,24)(26,28,32,30)(33,34)(35,45)(36,39)(37,46)(38,42)(40,44)(41,48)(43,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,18,22,25)(19,23,27,20)(21,24,31,29)(26,30,32,28)(33,35)(34,41)(36,46)(37,42)(38,43)(39,47)(40,48)(44,45);(2,7)(6,13)(8,11)(14,15)(17,19,22,27)(18,20,25,23)(21,26,31,32)(24,28,29,30)(33,36,40,38)(34,42,44,39)(35,46,48,43)(37,45,47,41);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,29)(31,32)(33,37,40,47)(34,43,44,46)(35,42,48,39)(36,41,38,45);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26)(33,38,40,36)(34,42,44,39)(35,43,48,46)(37,45,47,41);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,21)(18,24)(19,26)(20,28)(22,31)(23,30)(25,29)(27,32)(33,39,40,42)(34,38,44,36)(35,47,48,37)(41,43,45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,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=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9757794);