#8137629

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	#49851	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4159 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)(3,8)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30)(37,45)(40,47)(43,46)(44,48);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(18,23)(19,26)(24,30)(27,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,35,38,39)(34,36,41,42)(37,43,45,46)(40,44,47,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,46)(43,47)(45,48);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|c^2=hi,d^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,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, 8044825);