#128891

order	512
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

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

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, 36087);