#94499

order	512
nilpotency class	5
isoclinism	isoclinic class of #94438
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1393
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₂×#382	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#382 ◁ G
LECS	G ▷ C₂×#382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

272 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=i,e^2=f,g^2=i,b^a=bc,c^a=cf,e^a=egh,c^b=cg,d^b=dfh,g^b=gh,h^b=hi,e^c=ehi,e^d=ei,g^e=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1695);