#8505276

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	#53044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412472);