#140998

order	512
nilpotency class	4
isoclinism	isoclinic class of #125409
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂×#493	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=egi,d^2=ghi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,f^d=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, 48194);