#8746079

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

3954 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=i,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8653275);