#138221

order	512
nilpotency class	4
isoclinism	isoclinic class of #125425
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1555
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

397 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=h,d^2=gh,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45417);