#8107608

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	#40674	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014804);