#8533577

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

3969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=g,e^2=gh,f^2=gh,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, 8440773);