#8762068

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8669264);