#8818173

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	#46447	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8725369);