#417107

order	512
nilpotency class	3
isoclinism	isoclinic class of #402446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

716 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)(5,6,10,27,11,15,24,16)(7,17,13,22,20,12,19,8)(9,23,25,14,26,35,36,30)(18,32,29,28,34,21,33,31);(2,4)(5,7,10,19,11,20,24,13)(6,12,27,22,15,17,16,8)(9,18,25,33,26,34,36,29)(14,28,35,32,30,31,23,21);(1,2,3,4)(5,8,10,17,11,22,24,12)(6,13,16,7,15,19,27,20)(9,21,25,31,26,32,36,28)(14,29,23,18,30,33,35,34);(5,9,24,36,11,26,10,25)(6,14,27,35,15,30,16,23)(7,18,13,29,20,34,19,33)(8,21,12,28,22,32,17,31)

Polycyclic presentation:

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

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, 324303);