#444616

order	512
nilpotency class	3
isoclinism	isoclinic class of #417903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1078 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

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

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, 351812);