#10422711

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	#41268	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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)(3,5)(4,8)(6,9)(7,10)(11,13)(12,15)(14,16)(17,18,22,25)(19,23,28,21)(20,24,30,32)(26,31,27,29)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(1,3,7,4)(2,8,10,5)(6,11,14,12)(9,15,16,13)(17,19,22,28)(18,23,25,21)(20,27,30,26)(24,29,32,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,3,7,4)(2,5,10,8)(6,11,14,12)(9,13,16,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,36)(34,40)(35,37)(38,45)(39,41)(42,47)(43,48)(44,46);(1,4,7,3)(2,5,10,8)(6,11,14,12)(9,15,16,13)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,36)(34,40)(35,44)(37,46)(38,45)(39,48)(41,43)(42,47);(1,5)(2,4)(3,10)(6,13)(7,8)(9,12)(11,16)(14,15)(17,19)(18,21)(20,26)(22,28)(23,25)(24,29)(27,30)(31,32)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40);(1,6,7,14)(2,9,10,16)(3,11,4,12)(5,13,8,15)(17,21)(18,19)(20,29)(22,23)(24,26)(25,28)(27,32)(30,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=gh,e^2=h,f^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, 10329907);