#8475794

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4174 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8382990);