#8720128

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627324);