#147789

order	512
nilpotency class	4
isoclinism	isoclinic class of #129627
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×Q₈	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1065 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1065 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)(7,17)(8,20)(9,14,24,33)(10,25)(11,16)(12,18,30,21)(13,22)(15,27)(19,32)(23,28,26,31)(29,35,36,34);(1,2)(3,4)(5,7,11,8)(6,12,16,30)(9,21,24,18)(10,22,27,19)(13,31,32,28)(14,20,33,17)(15,29,25,36)(23,34,26,35);(2,4)(5,8,11,7)(6,13,16,32)(9,18,24,21)(10,19,27,22)(12,28,30,31)(14,29,33,36)(15,17,25,20)(23,35,26,34)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=ei,e^d=ei,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, 54985);