#8165154

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	#53051	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8072350);