#8720412

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	#38660	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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)(3,5)(4,12)(6,9)(7,11)(8,13)(10,16)(14,15)(17,18,23,24)(19,22,25,32)(20,21,29,30)(26,31,28,27)(33,34,38,42)(35,37,41,39)(36,44,46,40)(43,45,48,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,16)(12,14)(13,15)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,45,44,47);(4,13)(8,12)(10,15)(14,16)(18,24)(19,25)(20,29)(27,31)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,45,44,47);(1,4,6,13)(2,8,9,12)(3,10,11,15)(5,14,7,16)(17,20)(18,21)(19,26)(22,31)(23,29)(24,30)(25,28)(27,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,45)(44,47)(46,48);(1,5,6,7)(2,3,9,11)(4,14,13,16)(8,10,12,15)(17,21)(18,20)(19,27)(22,28)(23,30)(24,29)(25,31)(26,32)(33,36,38,46)(34,40,42,44)(35,43,41,48)(37,47,39,45);(1,3)(2,5)(4,15)(6,11)(7,9)(8,16)(10,13)(12,14)(17,22,23,32)(18,25,24,19)(20,27,29,31)(21,26,30,28)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,43,44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627608);