#8695956

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	#40276	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,14)(15,16)(18,22)(19,25)(24,27)(26,28)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(2,7)(3,12)(9,14)(10,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(34,39)(35,42)(41,44)(43,48);(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,3)(2,8)(4,10)(5,14)(6,12)(7,11)(9,15)(13,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4)(2,9)(3,10)(5,11)(6,13)(7,14)(8,15)(12,16)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5,6,15)(2,10,7,16)(3,9,12,14)(4,8,13,11)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32)(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=i,e^2=h,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, 8603152);