#499479

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3184
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23682
prime radical	#13340	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 573 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fi,g^c=gh,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, 406675);