#8127534

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=h,e^2=gh,f^2=g,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^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, 8034730);