#8750545

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,e^2=h,f^2=gi,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, 8657741);