#423850

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

1339 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)(5,6,9,15,11,16,10,12)(7,17,20,28,18,25,21,13)(8,14,22,29,24,30,23,26)(19,31,34,36,32,35,33,27);(6,12)(7,18)(8,22)(9,10)(13,25)(15,16)(17,28)(19,33)(23,24)(26,29)(31,35)(32,34);(5,7,10,21,11,18,9,20)(6,13,15,17,16,28,12,25)(8,19,23,33,24,32,22,34)(14,27,29,31,30,36,26,35);(2,4)(5,8,9,22,11,24,10,23)(6,14,15,29,16,30,12,26)(7,19,20,34,18,32,21,33)(13,27,17,31,28,36,25,35)

Polycyclic presentation:

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

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, 331046);