#457593

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	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1512 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=f,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, 364789);