#8522183

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

4316 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8429379);