#404530

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311726);