#8137113

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	#57855	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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)(3,8)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(20,28)(24,31)(26,32)(29,30)(34,37)(35,41)(36,44)(46,48);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,34,38,37)(35,39,41,43)(36,45,44,40)(42,48,47,46);(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,44)(34,40,37,45)(35,42,41,47)(39,46,43,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48)

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=h,f^2=gh,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, 8044309);