#423414

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
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

1344 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)(7,16,19,23)(12,25,27,17)(18,31,34,35)(26,33,36,32);(5,6)(7,17,19,25)(8,13)(9,22)(10,14)(11,15)(12,16,27,23)(18,33,34,32)(20,29)(21,30)(24,28)(26,35,36,31);(5,7,9,23,11,19,10,16)(6,12,14,17,15,27,22,25)(8,18,20,35,21,34,24,31)(13,26,28,32,30,36,29,33);(2,4)(5,8,11,21)(6,13,15,30)(7,18,19,34)(9,20,10,24)(12,26,27,36)(14,28,22,29)(16,31,23,35)(17,32,25,33)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,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, 330610);