#8533588

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	#57438	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) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,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, 8440784);