#8127527

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	#57962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,39,43)(35,44,46,40)(36,37,47,41)(38,48,45,42);(1,3,7,5)(2,8,4,11)(6,10,14,13)(9,15,12,16)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(33,35)(34,40)(36,38)(37,42)(39,46)(41,48)(43,44)(45,47);(2,4)(3,5)(9,12)(10,13)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36,39,47)(34,41,43,37)(35,38,46,45)(40,48,44,42);(1,4,7,2)(3,8,5,11)(6,12,14,9)(10,15,13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36)(34,41)(35,45)(37,43)(38,46)(39,47)(40,42)(44,48);(1,5,7,3)(2,8,4,11)(6,13,14,10)(9,15,12,16)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,37,39,41)(34,36,43,47)(35,42,46,48)(38,44,45,40);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,38,39,45)(34,42,43,48)(35,36,46,47)(37,44,41,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=gh,e^2=h,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, 8034723);