#357285

order	512
nilpotency class	3
isoclinism	isoclinic class of #357285
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13827
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1719 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 1719 conjugacy classes of subgroups — too many to draw. Only its 312 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)(3,11,14,7)(4,8)(5,18,19,9)(6,10)(12,20,15,25)(13,24)(16,29,30,22)(17,23)(21,27)(26,31)(28,32);(1,3)(2,7)(4,15,17,12)(5,13)(6,14)(8,20,23,25)(9,24)(10,11)(16,28,30,26)(18,21)(19,27)(22,32,29,31);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,23)(11,25)(13,26)(14,15)(18,29)(19,30)(21,31)(24,32)(27,28);(1,5)(2,9)(3,13)(4,16)(6,19)(7,21)(8,22)(10,18)(11,24)(12,26)(14,27)(15,28)(17,30)(20,31)(23,29)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₂×#72210029, #72263265
maximal subgroup of	#72210026, #72210029, #72210037, …
maximal normal subgroup of	#72210026, #72210029, #72210037, …
maximal quotient of	#72228237, #72238873, #72239046, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264481);