#8471142

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	#52556	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=i,e^2=gi,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, 8378338);