#9847626

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3743 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9754822);