#423852

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1333 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)(5,6,10,14,11,16,27,15)(7,17,9,25,19,29,26,12)(8,20,23,13,24,30,36,31)(18,28,22,32,21,34,35,33);(5,7,11,19)(6,12,16,25)(8,21,24,18)(9,10,26,27)(13,28,31,34)(14,29,15,17)(20,32,30,33)(22,36,35,23);(5,8,10,23,11,24,27,36)(6,13,15,20,16,31,14,30)(7,18,9,22,19,21,26,35)(12,28,29,33,25,34,17,32);(2,4)(5,9,10,19,11,26,27,7)(6,12,14,17,16,25,15,29)(8,22,23,21,24,35,36,18)(13,28,30,32,31,34,20,33)

Polycyclic presentation:

2^9|a^2=e,b^2=i,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, 331048);