#8165159

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	#53063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8072355);