#121072

order	512
nilpotency class	3
isoclinism	isoclinic class of #99293
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#383
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

408 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=d,d^2=hi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,g^b=gi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28268);