#8746555

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3807 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,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, 8653751);