#423408

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1378 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)(7,16,19,30)(8,13)(9,14)(10,23)(11,15)(12,20,26,17)(18,31,33,36)(21,27)(22,28)(24,29)(25,34,35,32);(7,17,19,20)(8,21)(12,16,26,30)(13,27)(18,32,33,34)(22,24)(25,31,35,36)(28,29);(5,7,9,17,11,19,10,20)(6,12,14,30,15,26,23,16)(8,18,22,32,21,33,24,34)(13,25,28,36,27,35,29,31);(2,4)(5,8)(6,13)(7,18)(9,22)(10,24)(11,21)(12,25)(14,28)(15,27)(16,31)(17,32)(19,33)(20,34)(23,29)(26,35)(30,36)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,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, 330604);