#422685

order	512
nilpotency class	3
isoclinism	isoclinic class of #418508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

1062 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)(5,6,11,15)(7,16,19,31)(8,20,24,36)(9,25,18,12)(10,27,26,14)(13,29,30,23)(17,33,21,28)(22,35,34,32);(1,3,4,2)(5,7,11,19)(6,12,15,25)(8,21,24,17)(9,26,18,10)(13,28,30,33)(14,31,27,16)(20,35,36,32)(22,23,34,29);(2,3)(5,8,10,23,11,24,26,29)(6,13,14,20,15,30,27,36)(7,17,18,34,19,21,9,22)(12,28,16,32,25,33,31,35);(5,9,10,7,11,18,26,19)(6,12,14,16,15,25,27,31)(8,22,23,17,24,34,29,21)(13,28,20,32,30,33,36,35)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 329881);