#8359971

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	#56339	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8267167);