#8745775

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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)(3,5,4,11)(6,9,15,16)(10,13,12,14)(17,18)(19,26)(20,22)(21,25)(23,24)(27,28)(29,32)(30,31)(35,43)(36,45)(39,46)(40,48);(1,3)(2,5)(4,7)(6,10)(8,11)(9,13)(12,15)(14,16)(18,24)(20,29)(25,26)(27,30)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,44,47);(2,8)(3,4)(9,16)(10,12)(17,19,23,21)(18,25,24,26)(20,30,29,27)(22,28,32,31)(33,35,38,43)(34,39,42,46)(36,44,45,37)(40,47,48,41);(1,4,7,3)(2,5,8,11)(6,10,15,12)(9,14,16,13)(17,20,23,29)(18,22,24,32)(19,27,21,30)(25,31,26,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5)(2,3)(4,8)(6,14)(7,11)(9,12)(10,16)(13,15)(17,21,23,19)(18,26,24,25)(20,30,29,27)(22,28,32,31)(33,36)(34,40)(35,44)(37,43)(38,45)(39,47)(41,46)(42,48);(1,6,7,15)(2,9,8,16)(3,10,4,12)(5,13,11,14)(17,22)(18,20)(19,28)(21,31)(23,32)(24,29)(25,30)(26,27)(33,37,38,44)(34,41,42,47)(35,36,43,45)(39,40,46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8652971);