#8733924

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	#58883	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3955 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8641120);