#263408

order	512
nilpotency class	3
isoclinism	isoclinic class of #263403
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18696	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=ei,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170604);