#423844

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1300 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 331040);