#8115385

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3932 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)(3,6)(4,12)(5,9)(7,11)(8,13)(10,15)(14,16)(17,18)(19,25)(20,24)(21,29)(22,32)(23,26)(27,30)(28,31)(33,34,39,41)(35,40,36,44)(37,38,47,48)(42,46,45,43);(1,3,5,11)(2,6,9,7)(4,10,13,16)(8,14,12,15)(17,19,23,29)(18,21,26,25)(20,27,22,28)(24,31,32,30)(33,35,39,36)(34,40,41,44)(37,45,47,42)(38,43,48,46);(1,2,5,9)(3,7,11,6)(4,8,13,12)(10,15,16,14)(17,20,23,22)(18,24,26,32)(19,27,29,28)(21,31,25,30)(33,36,39,35)(34,40,41,44)(37,45,47,42)(38,46,48,43);(1,2)(3,7)(4,8)(5,9)(6,11)(10,15)(12,13)(14,16)(17,21,23,25)(18,19,26,29)(20,30,22,31)(24,28,32,27)(33,37)(34,38)(35,42)(36,45)(39,47)(40,46)(41,48)(43,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,19)(18,25)(20,27)(21,26)(22,28)(23,29)(24,30)(31,32)(33,38)(34,37)(35,43)(36,46)(39,48)(40,45)(41,47)(42,44);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,22,23,20)(18,24,26,32)(19,28,29,27)(21,31,25,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022581);