#138024

order	512
nilpotency class	4
isoclinism	isoclinic class of #125094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

424 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(6,9,10,11)(13,14,18,25,20,26,19,24)(15,27,30,37,32,43,31,42)(16,23,17,22,34,36,33,41)(21,35,39,44,40,28,38,29);(5,6,7,11,8,10,12,9)(13,15,17,29,18,30,34,35,20,32,33,44,19,31,16,28)(14,21,36,43,24,38,22,37,26,40,23,27,25,39,41,42);(2,4)(13,16,19,33,20,34,18,17)(14,22,25,36,26,41,24,23)(15,28,31,44,32,35,30,29)(21,37,39,43,40,42,38,27)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=ghi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,f^b=fh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45220);