#10407761

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	#37348	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3951 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10314957);