#8739380

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	#52665	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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)(3,5)(4,11)(6,7)(8,13)(9,15)(10,12)(14,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,32)(30,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,7)(3,9)(8,11)(10,16)(17,19,22,28)(18,23,26,27)(20,25,29,30)(21,24,31,32)(34,35)(39,41)(40,42)(46,47);(2,7)(3,9)(8,11)(10,16)(17,20,22,29)(18,24,26,32)(19,25,28,30)(21,23,31,27)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,3,6,9)(2,5,7,15)(4,10,13,16)(8,14,11,12)(17,20)(18,24)(19,25)(21,27)(22,29)(23,31)(26,32)(28,30)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4,6,13)(2,8,7,11)(3,10,9,16)(5,14,15,12)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32);(1,5)(2,3)(4,12)(6,15)(7,9)(8,16)(10,11)(13,14)(17,21,22,31)(18,25,26,30)(19,24,28,32)(20,23,29,27)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=h,e^2=h,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, 8646576);