#8159041

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	#56292	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 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)(3,4)(5,7)(6,13)(8,10)(9,14)(11,16)(12,15)(17,18)(19,29)(20,26)(21,22)(23,31)(24,28)(25,32)(27,30)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(1,3,7,8)(2,4,5,10)(6,11,14,15)(9,12,13,16)(17,19)(18,25)(20,27)(21,30)(22,31)(23,26)(24,32)(28,29)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11)(17,20,24,22)(18,26,28,21)(19,27,32,31)(23,29,30,25)(33,34,38,40)(35,36,43,41)(37,39,46,45)(42,44,48,47);(1,4)(2,3)(5,8)(6,12)(7,10)(9,11)(13,15)(14,16)(17,21,24,26)(18,20,28,22)(19,30,32,23)(25,27,29,31)(33,36,38,41)(34,35,40,43)(37,44,46,47)(39,42,45,48);(1,5,7,2)(3,10,8,4)(6,13,14,9)(11,16,15,12)(17,22,24,20)(18,21,28,26)(19,31,32,27)(23,25,30,29)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(1,6,7,14)(2,9,5,13)(3,11,8,15)(4,12,10,16)(17,23,24,30)(18,27,28,31)(19,26,32,21)(20,29,22,25)(33,37)(34,39)(35,42)(36,44)(38,46)(40,45)(41,47)(43,48)

Polycyclic presentation:

2^9|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, 8066237);