#418857

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4)(6,12)(7,17)(9,10)(13,18)(14,27)(15,16)(19,33)(20,21)(22,30)(23,24)(31,35)(34,36);(5,6,11,16)(7,18,21,13)(8,22,25,27)(9,15,10,12)(14,24,30,23)(17,26,20,29)(19,34,35,36)(28,31,32,33);(2,4)(5,7)(6,13)(8,19)(9,17)(10,20)(11,21)(12,26)(14,28)(15,29)(16,18)(22,34)(23,33)(24,31)(25,35)(27,36)(30,32);(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,27)(13,28)(15,22)(16,30)(17,31)(18,32)(20,33)(21,35)(26,36)(29,34)

Polycyclic presentation:

2^9|a^2=ef,b^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326053);