#93518

order	512
nilpotency class	4
isoclinism	isoclinic class of #93517
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#340
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂⁴×C₈	C₂²
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂⁴×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

578 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 714);