#8127387

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3928 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=g,e^2=h,f^2=g,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^c=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, 8034583);