#423418

order	512
nilpotency class	3
isoclinism	isoclinic class of #419461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 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)(7,15,17,21)(11,23,25,28)(16,29,31,33)(24,35,30,36);(5,6)(7,11,17,25)(8,12)(9,20)(10,14)(13,22)(15,28,21,23)(16,30,31,24)(18,32)(19,27)(26,34)(29,35,33,36);(5,7,9,21,10,17,22,15)(6,11,13,28,14,25,20,23)(8,16,18,33,19,31,34,29)(12,24,26,36,27,30,32,35);(2,4)(5,8,10,19)(6,12,14,27)(7,16,17,31)(9,18,22,34)(11,24,25,30)(13,26,20,32)(15,29,21,33)(23,35,28,36)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=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, 330614);