#8662950

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	#53435	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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)(3,9,14,8)(4,6,5,15)(11,16,12,13)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,34)(35,39)(36,45)(37,40)(38,42)(41,44)(43,48)(46,47);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,10)(15,16)(18,20)(23,25)(24,28)(31,32)(33,35)(34,39)(36,46)(37,43)(38,44)(40,48)(41,42)(45,47);(1,4,7,5)(2,6,10,15)(3,11,14,12)(8,13,9,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,36,38,37)(34,40,42,45)(35,43,44,46)(39,47,41,48);(1,5,7,4)(2,6,10,15)(3,12,14,11)(8,13,9,16)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,34)(35,41)(36,40)(37,45)(38,42)(39,44)(43,48)(46,47);(1,3)(2,9)(4,11)(5,12)(6,16)(7,14)(8,10)(13,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,35)(34,41)(36,43)(37,46)(38,44)(39,42)(40,48)(45,47);(1,6)(2,5)(3,13)(4,10)(7,15)(8,12)(9,11)(14,16)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37,38,36)(34,40,42,45)(35,43,44,46)(39,48,41,47)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570146);