#8463658

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	#42448	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,e^2=ghi,f^2=hi,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, 8370854);