#8695272

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	#51910	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(17,18)(19,25)(20,21)(22,26)(23,24)(27,30)(28,29)(31,32)(34,36)(35,41)(40,44)(43,45);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(18,24)(20,25)(26,32)(28,30)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,30,32,28)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,20)(18,19)(21,24)(22,30)(23,25)(26,29)(27,32)(28,31)(33,36,38,34)(35,42,41,39)(37,40,46,44)(43,47,45,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,21,23,19)(18,20,24,25)(22,27,31,29)(26,30,32,28)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,37,38,46)(34,40,36,44)(35,43,41,45)(39,47,42,48)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8602468);