#94651

order	512
nilpotency class	6
isoclinism	isoclinic class of #94650
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7403
commutator	C₄³	C₂×C₄
Frattini, cosocle	#1962	C₂²
prime radical	#532	D₄
LCS	G ▷ C₄³ ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1962 ◁ G
LECS	G ▷ #1962 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

178 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 24 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=hi,c^2=fi,e^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,h^a=hi,c^b=cf,d^b=defhi,e^b=eh,g^b=gi,d^c=dghi,e^d=ehi,f^d=fi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1847);