#465045

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

980 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 372241);