#8581796

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	#52419	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8488992);