#8096583

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

4026 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)(3,10,8,4)(6,9,14,13)(11,16,15,12)(17,18)(19,27)(20,25)(21,26)(22,23)(24,28)(29,31)(30,32)(33,34,38,42)(35,37,41,39)(36,40,44,47)(43,45,48,46);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,16)(14,15)(18,23)(19,28)(25,32)(26,29)(36,44)(40,47)(43,48)(45,46);(17,19)(18,24)(20,29)(21,25)(22,28)(23,27)(26,30)(31,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,48)(45,47);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11)(17,19)(18,24)(20,29)(21,25)(22,28)(23,27)(26,30)(31,32)(33,36,38,44)(34,40,42,47)(35,43,41,48)(37,45,39,46);(1,5,7,2)(3,10,8,4)(6,9,14,13)(11,12,15,16)(17,20)(18,25)(19,29)(21,24)(22,30)(23,32)(26,28)(27,31)(33,35)(34,37)(36,43)(38,41)(39,42)(40,45)(44,48)(46,47);(1,6,7,14)(2,9,5,13)(3,11,8,15)(4,12,10,16)(17,21)(18,26)(19,25)(20,27)(22,31)(23,29)(24,30)(28,32)(33,37)(34,41)(35,42)(36,45)(38,39)(40,48)(43,47)(44,46)

Polycyclic presentation:

2^9|a^2=gh,d^2=gh,e^2=gi,f^2=gi,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, 8003779);