#152779

order	512
nilpotency class	5
isoclinism	isoclinic class of #152778
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13090
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#3500	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

503 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59975);