#8159105

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	#40368	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3924 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)(3,13)(4,10)(5,6)(7,15)(8,12)(9,16)(11,14)(17,18,22,25)(19,26,23,20)(21,29,30,24)(27,28,31,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(1,3)(2,9)(4,11)(5,14)(6,15)(7,10)(8,16)(12,13)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,34,38,42)(35,39,46,43)(36,40,37,41)(44,47,45,48);(1,4,8,6)(2,10,12,5)(3,11,16,15)(7,13,14,9)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,8,10)(2,4,12,6)(3,14,16,7)(9,11,13,15)(17,20,22,26)(18,23,25,19)(21,28,30,32)(24,31,29,27)(33,35,38,46)(34,39,42,43)(36,44,37,45)(40,47,41,48);(1,6,8,4)(2,5,12,10)(3,15,16,11)(7,9,14,13)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(1,7,8,14)(2,11,12,15)(3,10,16,5)(4,13,6,9)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=g,e^2=gh,f^2=gh,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, 8066301);