#413200

order	512
nilpotency class	3
isoclinism	isoclinic class of #397950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60639	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320396);