#8097085

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4040 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=gh,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, 8004281);