#457230

order	512
nilpotency class	3
isoclinism	isoclinic class of #454369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

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

1321 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,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, 364426);