#8755075

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	#58364	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=gi,e^2=h,f^2=g,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, 8662271);