#8818134

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	#46613	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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:

(17,18,24,28)(19,26,21,22)(20,25,23,27)(29,30,31,32)(34,38)(36,43)(39,44)(42,47);(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)(35,41)(39,44)(42,47)(46,48);(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)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(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)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(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)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 8725330);