#8463674

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	#51967	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=g,e^2=hi,f^2=ghi,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, 8370870);