#8979229

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4065 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,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, 8886425);