#457598

order	512
nilpotency class	3
isoclinism	isoclinic class of #454002
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1534 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=f,f^2=i,g^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,g^b=gi,d^c=de,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, 364794);