#152127

order	512
nilpotency class	5
isoclinism	isoclinic class of #152127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12698
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3444 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

565 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fgh,g^b=gi,f^c=fi,f^d=fhi,g^f=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59323);