#8136907

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	#55222	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044103);