#8463659

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,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, 8370855);