#10408566

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	#2846	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gh,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, 10315762);