#430458

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1191 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=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, 337654);