#8165162

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	#57484	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(3,9)(4,5)(6,14)(7,8)(10,15)(11,12)(13,16)(17,18,21,25)(19,23,26,22)(20,28,29,24)(27,30,32,31)(33,34)(35,41)(36,42)(37,40)(38,39)(43,47)(44,48)(45,46);(1,3,7,5)(2,4,8,9)(6,11,15,13)(10,12,14,16)(19,26)(22,23)(27,32)(30,31)(34,39)(35,42)(40,46)(43,48);(2,8)(4,9)(10,14)(12,16)(17,19,21,26)(18,22,25,23)(20,27,29,32)(24,30,28,31)(33,35)(34,36)(37,43)(38,42)(39,41)(40,44)(45,48)(46,47);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,14)(15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,35)(34,36)(37,43)(38,42)(39,41)(40,44)(45,48)(46,47);(1,5,7,3)(2,9,8,4)(6,13,15,11)(10,16,14,12)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,6,7,15)(2,10,8,14)(3,11,5,13)(4,12,9,16)(17,20,21,29)(18,24,25,28)(19,27,26,32)(22,30,23,31)(33,37,38,45)(34,40,39,46)(35,43,42,48)(36,44,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,e^2=hi,f^2=h,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, 8072358);