#10410040

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	#50849	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10317236);