#8436631

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	#45556	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343827);