#460304

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 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)(3,4)(5,6,9,14,10,11,22,24)(7,15,16,25,18,12,23,27)(8,19,20,33,21,13,28,30)(17,29,31,35,32,26,34,36);(6,11)(7,16)(9,22)(12,25)(13,19)(15,27)(17,31)(18,23)(20,28)(26,35)(29,36)(32,34);(2,4)(5,7)(6,12)(8,17)(9,23)(10,18)(11,15)(13,29)(14,25)(16,22)(19,26)(20,34)(21,32)(24,27)(28,31)(30,36)(33,35);(5,8,10,21)(6,13)(7,17,18,32)(9,20,22,28)(11,19)(12,26)(14,30)(15,29)(16,31,23,34)(24,33)(25,35)(27,36)

Polycyclic presentation:

2^9|a^2=g,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,g^c=gi,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, 367500);