#8467053

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 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)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18,23,22)(19,24,29,27)(20,25,30,32)(21,31,28,26)(33,34)(35,41)(36,43)(37,40)(38,39)(42,45)(44,48)(46,47);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(17,18,23,22)(19,27,29,24)(20,25,30,32)(21,26,28,31)(34,39)(36,41)(40,47)(44,45);(17,19,23,29)(18,24,22,27)(20,28,30,21)(25,26,32,31)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,20)(18,25)(19,28)(21,29)(22,32)(23,30)(24,26)(27,31)(33,36)(34,35)(37,45)(38,41)(39,43)(40,48)(42,47)(44,46);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,21,23,28)(18,26,22,31)(19,20,29,30)(24,32,27,25);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,22,23,18)(19,27,29,24)(20,25,30,32)(21,31,28,26)(33,37,38,46)(34,40,39,47)(35,42,43,48)(36,44,41,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=gh,e^2=ghi,f^2=gi,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, 8374249);