#97062

order	512
nilpotency class	3
isoclinism	isoclinic class of #97058
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

1014 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,e^b=eh,f^b=fg,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4258);