#206054

order	512
nilpotency class	3
isoclinism	isoclinic class of #205765
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3191	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113250);