#499633

order	512
nilpotency class	4
isoclinism	isoclinic class of #499262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3184
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23781
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

435 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406829);