#8384229

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	#47993	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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:

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

Polycyclic presentation:

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

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, 8291425);