#446128

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353324);