#8097136

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(3,10)(4,12)(5,8)(6,9)(7,14)(11,13)(15,16)(17,18)(19,27)(20,29)(21,23)(22,25)(24,26)(28,31)(30,32)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,3,6,8)(2,5,9,10)(4,11,14,15)(7,13,12,16)(17,19,24,23)(18,21,26,27)(20,30,22,28)(25,32,29,31)(35,41)(39,42)(43,47)(46,48);(17,20,24,22)(18,25,26,29)(19,28,23,30)(21,31,27,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4)(2,7)(3,11)(5,13)(6,14)(8,15)(9,12)(10,16)(17,21)(18,19)(20,31)(22,32)(23,26)(24,27)(25,28)(29,30);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,14)(15,16)(17,22,24,20)(18,25,26,29)(19,28,23,30)(21,32,27,31)(33,36,38,34)(35,42,41,39)(37,40,45,44)(43,46,47,48);(1,5)(2,8)(3,9)(4,13)(6,10)(7,15)(11,12)(14,16)(17,23,24,19)(18,21,26,27)(20,30,22,28)(25,31,29,32)(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=g,c^2=h,e^2=i,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, 8004332);