#422227

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1040 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)(5,6,12,18,11,17,10,16)(7,15,21,13,20,34,9,28)(8,22,27,32,26,14,25,33)(19,30,23,36,35,31,24,29);(1,3,4,2)(5,7,11,20)(6,13,17,28)(8,23,26,24)(9,10,21,12)(14,30,22,31)(15,16,34,18)(19,25,35,27)(29,32,36,33);(2,3)(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,27)(13,29)(15,31)(16,32)(17,22)(18,33)(20,35)(21,23)(28,36)(30,34);(5,9,10,20,11,21,12,7)(6,15,16,28,17,34,18,13)(8,24,25,35,26,23,27,19)(14,31,32,36,22,30,33,29)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,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, 329423);