#434074

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1187 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)(5,6,12,15,11,17,10,16)(7,18,20,23,21,29,13,8)(9,24,28,33,27,14,26,32)(19,31,30,36,35,34,25,22);(5,7,10,13,11,21,12,20)(6,8,15,18,17,23,16,29)(9,25,26,35,27,30,28,19)(14,31,32,36,24,34,33,22);(5,8)(6,13,17,20)(7,15,21,16)(9,22)(10,18)(11,23)(12,29)(14,30,24,25)(19,33,35,32)(26,31)(27,36)(28,34);(2,4)(5,9,11,27)(6,14,17,24)(7,19,21,35)(8,22,23,36)(10,28,12,26)(13,30,20,25)(15,33,16,32)(18,34,29,31)

Polycyclic presentation:

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

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, 341270);