#127768

order	512
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

385 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=g,f^2=ghi,h^2=i,b^a=bd,d^a=dg,f^a=fgh,h^a=hi,c^b=ch,d^b=dh,g^b=gi,h^b=hi,d^c=di

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, 34964);