#8762072

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,e^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, 8669268);