#457090

order	512
nilpotency class	3
isoclinism	isoclinic class of #457090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1131 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^c=hi,h^d=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, 364286);