#147600

order	512
nilpotency class	4
isoclinism	isoclinic class of #129431
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

931 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 931 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54796);