#432004

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

1196 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)(5,6,9,15,11,17,10,16)(7,18,21,29,22,33,27,12)(8,14,24,13,26,30,25,31)(19,28,20,32,23,36,35,34);(5,7,10,21,11,22,9,27)(6,12,16,18,17,29,15,33)(8,23,25,35,26,19,24,20)(13,28,14,32,31,36,30,34);(5,8)(6,13)(7,19,22,23)(9,25)(10,24)(11,26)(12,28,29,36)(14,15)(16,30)(17,31)(18,34,33,32)(20,21,35,27);(2,4)(5,8)(6,14)(7,20,22,35)(9,24)(10,25)(11,26)(12,28,29,36)(13,15)(16,31)(17,30)(18,32,33,34)(19,21,23,27)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ef,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 339200);