#8438068

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	#53585	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4108 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8345264);