#413122

order	512
nilpotency class	3
isoclinism	isoclinic class of #397258
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1038 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 234 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)(5,6,11,15,10,14,9,25)(7,16,18,12,20,29,19,28)(8,21,23,31,24,13,26,35)(17,32,22,36,34,30,33,27);(5,7,11,18,10,20,9,19)(6,12,15,29,14,28,25,16)(8,22,23,34,24,33,26,17)(13,30,35,27,21,32,31,36);(5,8,9,23,10,24,11,26)(6,13,15,31,14,21,25,35)(7,17,18,33,20,34,19,22)(12,27,16,32,28,36,29,30);(1,2,3,4)(6,14)(12,28)(13,21)(15,25)(16,29)(27,36)(30,32)(31,35)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,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, 320318);