#417962

order	512
nilpotency class	3
isoclinism	isoclinic class of #417566
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

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

2064 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)(14,24)(15,31,32,25)(16,33)(17,26)(19,28,34,36)(21,29,23,30)(27,35);(5,7,10,17)(6,11,13,26)(8,19)(9,16,24,35)(12,28)(14,27,22,33)(15,23)(18,36)(20,34)(21,32)(25,30)(29,31);(5,8)(6,12)(7,15)(9,23)(10,20)(11,25)(13,18)(14,30)(16,34)(17,32)(19,35)(21,24)(22,29)(26,31)(27,36)(28,33);(2,4)(9,24)(14,22)(16,35)(19,34)(21,23)(27,33)(28,36)(29,30)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,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, 325158);