#415236

order	512
nilpotency class	3
isoclinism	isoclinic class of #404632
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

648 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:

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

Polycyclic presentation:

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

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, 322432);