#8457037

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	#41297	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,e^2=i,f^2=h,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, 8364233);