#387631

order	512
nilpotency class	3
isoclinism	isoclinic class of #232138
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1058 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 242 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=efi,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi

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, 294827);