#8695828

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	#48075	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4043 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gh,e^2=g,f^2=g,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, 8603024);