#422505

order	512
nilpotency class	3
isoclinism	isoclinic class of #419041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1226 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)(5,6,10,15,12,9,11,26)(7,13,18,29,19,17,20,34)(8,21,24,23,25,31,27,14)(16,32,33,30,22,36,35,28);(2,3)(5,7,10,18,12,19,11,20)(6,13,15,29,9,17,26,34)(8,22,24,35,25,16,27,33)(14,30,21,36,23,28,31,32);(1,2,4,3)(5,8)(6,14)(7,16)(9,23)(10,27)(11,24)(12,25)(13,28)(15,31)(17,30)(18,35)(19,22)(20,33)(21,26)(29,36)(32,34);(1,3,4,2)(5,9,12,6)(7,17,19,13)(8,23,25,14)(10,26,11,15)(16,30,22,28)(18,34,20,29)(21,24,31,27)(32,33,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329701);