#8757207

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
prime radical	#45988	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664403);