#8621482

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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)(3,10)(4,12)(5,6)(7,8)(9,15)(11,14)(13,16)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32)(33,34,40,38)(35,39,43,41)(36,42,37,47)(44,46,45,48);(1,3,7,6)(2,5,8,10)(4,13,15,11)(9,16,12,14)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32)(33,35,40,43)(34,41,38,39)(36,44,37,45)(42,48,47,46);(2,8)(5,10)(9,12)(14,16)(19,24)(21,29)(27,30)(28,32)(33,36,40,37)(34,42,38,47)(35,44,43,45)(39,46,41,48);(1,4,7,15)(2,9,8,12)(3,11,6,13)(5,14,10,16)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32)(33,37,40,36)(34,42,38,47)(35,45,43,44)(39,46,41,48);(1,5)(2,3)(4,14)(6,8)(7,10)(9,11)(12,13)(15,16)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27)(33,38,40,34)(35,41,43,39)(36,42,37,47)(44,46,45,48);(1,6,7,3)(2,5,8,10)(4,11,15,13)(9,16,12,14)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)(33,39,40,41)(34,43,38,35)(36,46,37,48)(42,45,47,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8528678);