#151944

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2326 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

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

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, 59140);