#8114371

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	#58835	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=i,e^2=i,f^2=g,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, 8021567);