#434075

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1212 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)(5,6,9,16,10,15,22,14)(7,17,19,11,20,28,26,13)(8,12,25,30,24,31,23,32)(18,34,36,27,21,33,35,29);(5,7,9,19,10,20,22,26)(6,11,14,17,15,13,16,28)(8,21,23,36,24,18,25,35)(12,27,30,33,31,29,32,34);(5,8,22,25,10,24,9,23)(6,12,14,30,15,31,16,32)(7,18,19,35,20,21,26,36)(11,27,28,34,13,29,17,33);(2,4)(5,7)(6,13)(8,18)(9,26)(10,20)(11,15)(12,29)(14,17)(16,28)(19,22)(21,24)(23,36)(25,35)(27,31)(30,33)(32,34)

Polycyclic presentation:

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

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, 341271);