#8114379

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	#37228	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=i,e^2=gi,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, 8021575);