#8097222

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	#46298	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,10)(4,12)(5,7)(6,9)(8,14)(11,15)(13,16)(17,18)(19,20)(21,29)(22,24)(23,26)(25,32)(27,31)(28,30)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,3,6,5)(2,7,9,10)(4,11,14,13)(8,15,12,16)(17,19)(18,22)(20,26)(21,30)(23,24)(25,28)(27,32)(29,31)(35,41)(39,42)(43,47)(46,48);(17,20,23,22)(18,24,26,19)(21,28,32,31)(25,30,29,27)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,32)(24,30)(26,29);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,36,38,34)(35,42,41,39)(37,40,45,44)(43,46,47,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,22,23,20)(18,24,26,19)(21,31,32,28)(25,30,29,27)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004418);