#93229

order	512
nilpotency class	5
isoclinism	isoclinic class of #93229
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₂×#97	C₂×C₄
Frattini, cosocle	#1955	C₂²
socle	C₂²	#1343
prime radical	#10032	C₂
LCS	G ▷ C₂×#97 ▷ C₂×C₄² ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1955 ◁ G
LECS	G ▷ #1955 ▷ C₂×C₄² ▷ C₂³ ▷ C₂² ▷ C₁

227 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 26 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)(3,10,11,6)(4,8,14,7)(5,9)(12,15,13,16)(18,22)(19,25,28,29)(23,26)(24,32)(30,31);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14)(17,18)(19,26)(20,30)(21,24)(22,29)(23,27)(25,32)(28,31)

Polycyclic presentation:

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

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, 425);