#8324113

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	#40148	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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)(3,5)(4,12)(6,8)(7,11)(9,15)(10,16)(13,14)(17,18,23,26)(19,22,25,31)(20,21,24,30)(27,28,29,32)(33,34)(35,41)(36,38)(37,40)(39,43)(42,46)(44,48)(45,47);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,19)(18,22)(20,27)(21,28)(23,25)(24,29)(26,31)(30,32)(33,35,39,37)(34,40,43,41)(36,44,46,47)(38,45,42,48);(2,8)(5,7)(9,12)(14,16)(19,25)(22,31)(27,29)(28,32)(33,35,39,37)(34,41,43,40)(36,44,46,47)(38,48,42,45);(1,4,6,15)(2,9,8,12)(3,10,11,13)(5,14,7,16)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,30)(31,32)(33,36)(34,42)(35,44)(37,47)(38,43)(39,46)(40,48)(41,45);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,21)(18,24)(19,28)(20,26)(22,29)(23,30)(25,32)(27,31)(33,37,39,35)(34,40,43,41)(36,44,46,47)(38,48,42,45);(1,5,6,7)(2,3,8,11)(4,14,15,16)(9,10,12,13)(17,22,23,31)(18,25,26,19)(20,28,24,32)(21,29,30,27)(33,38,39,42)(34,36,43,46)(35,45,37,48)(40,44,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8231309);