#8115440

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

3850 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)(3,11)(4,15)(5,7)(6,16)(8,14)(9,13)(10,12)(17,18)(19,25)(20,24)(21,29)(22,32)(23,26)(27,30)(28,31)(33,34,39,42)(35,37,40,36)(38,46,47,41)(43,44,48,45);(1,3,7,14)(2,8,5,11)(4,6,13,12)(9,16,15,10)(17,19,23,29)(18,21,26,25)(20,27,22,28)(24,31,32,30)(33,35,39,40)(34,36,42,37)(38,43,47,48)(41,44,46,45);(1,2,7,5)(3,8,14,11)(4,15,13,9)(6,10,12,16)(17,20,23,22)(18,24,26,32)(19,27,29,28)(21,31,25,30)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,48,46,43);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,10,11,16)(17,21,23,25)(18,19,26,29)(20,30,22,31)(24,28,32,27)(33,37,39,36)(34,35,42,40)(38,45,47,44)(41,43,46,48);(1,5,7,2)(3,8,14,11)(4,15,13,9)(6,16,12,10)(17,19)(18,25)(20,27)(21,26)(22,28)(23,29)(24,30)(31,32)(33,38,39,47)(34,41,42,46)(35,43,40,48)(36,44,37,45);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,9,11,15)(17,22,23,20)(18,24,26,32)(19,28,29,27)(21,31,25,30)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=h,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, 8022636);