#8467223

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(17,18,22,26)(19,27,28,23)(20,24,29,32)(21,25,31,30)(33,34,39,38)(35,40,45,43)(36,41,46,48)(37,47,44,42);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,19)(18,23)(20,25)(21,24)(22,28)(26,27)(29,30)(31,32)(33,34,39,38)(35,43,45,40)(36,41,46,48)(37,42,44,47);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,28)(29,32)(30,31)(33,35,39,45)(34,40,38,43)(36,44,46,37)(41,42,48,47);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,18)(19,27)(20,24)(21,30)(22,26)(23,28)(25,31)(29,32)(33,36)(34,41)(35,44)(37,45)(38,48)(39,46)(40,42)(43,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,20,22,29)(18,24,26,32)(19,25,28,30)(21,27,31,23)(33,37,39,44)(34,42,38,47)(35,36,45,46)(40,48,43,41);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,21,22,31)(18,25,26,30)(19,24,28,32)(20,27,29,23)(33,38,39,34)(35,43,45,40)(36,41,46,48)(37,47,44,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=gh,e^2=hi,f^2=gi,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, 8374419);