#8695984

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	#55570	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=g,e^2=gh,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, 8603180);