#493404

order	512
nilpotency class	4
isoclinism	isoclinic class of #493397
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1781
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#62842	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

3441 conjugacy classes of subgroups; only the 328 normal subgroups are shown.

normal subgroups only

This group has 3441 conjugacy classes of subgroups — too many to draw. Only its 328 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,d^c=di,e^c=ei,g^e=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338768, #72338780, #72338802
maximal normal subgroup of	#72338768, #72338780, #72338802

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400600);