#457594

order	512
nilpotency class	3
isoclinism	isoclinic class of #457592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1544 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364790);