#8695877

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	#38509	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4177 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

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, 8603073);