#400058

order	512
nilpotency class	3
isoclinism	isoclinic class of #400057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1371 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=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, 307254);