#429602

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1047 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)(5,6,9,14,10,15,25,12)(7,16,17,28,20,27,19,11)(8,13,22,31,24,32,23,29)(18,30,21,35,34,36,33,26);(5,7,9,19,10,20,25,17)(6,11,14,27,15,28,12,16)(8,21,22,18,24,33,23,34)(13,30,31,26,32,36,29,35);(6,12)(7,17)(9,25)(13,29)(14,15)(16,27)(18,21)(19,20)(22,23)(30,36)(31,32)(33,34);(2,4)(5,8,10,24)(6,13,15,32)(7,18)(9,22,25,23)(11,26)(12,29,14,31)(16,30)(17,21)(19,33)(20,34)(27,36)(28,35)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,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, 336798);