#8107648

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014844);