#94619

order	512
nilpotency class	6
isoclinism	isoclinic class of #94618
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7394
commutator	#407	C₂×C₄
Frattini, cosocle	#1783	C₂²
prime radical	C₂³×D₄	D₄
LCS	G ▷ #407 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1783 ◁ G
LECS	G ▷ #1783 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 22 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=fi,f^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,h^a=hi,c^b=cf,d^b=def,f^b=fi,g^b=gh,d^c=dg,e^c=ehi,g^c=gi,f^d=fi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1815);