#8324098

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=h,e^2=h,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, 8231294);