#444617

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1118 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1118 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)(5,6,10,15,9,16,11,14)(7,12,19,27,18,28,20,32)(8,21,24,30,23,13,25,31)(17,33,22,36,34,26,35,29);(5,7,9,18)(6,12,16,28)(8,22,23,35)(10,20,11,19)(13,29,21,36)(14,27,15,32)(17,25,34,24)(26,30,33,31);(5,8,11,25,9,23,10,24)(6,13,15,31,16,21,14,30)(7,17,20,35,18,34,19,22)(12,26,27,29,28,33,32,36);(2,4)(6,14,16,15)(7,18)(12,27,28,32)(13,30,21,31)(17,34)(19,20)(22,35)(26,29,33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=eg,d^2=ei,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, 351813);