#449589

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2084 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)(7,11)(8,18,20,12)(9,22)(10,13)(14,25)(15,33,34,26)(16,31)(17,27)(19,29,35,36)(21,30,24,32)(23,28);(5,7,10,17)(6,11,13,27)(8,19,20,35)(9,23,25,16)(12,29,18,36)(14,31,22,28)(15,24,34,21)(26,32,33,30);(5,8)(6,12)(7,15)(9,24)(10,20)(11,26)(13,18)(14,32)(16,35)(17,34)(19,23)(21,25)(22,30)(27,33)(28,36)(29,31);(2,4)(9,25)(14,22)(16,23)(19,35)(21,24)(28,31)(29,36)(30,32)

Polycyclic presentation:

2^9|a^2=eh,b^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,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, 356785);