#141070

order	512
nilpotency class	4
isoclinism	isoclinic class of #125438
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1558
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,d^2=ghi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,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, 48266);