#93801

order	512
nilpotency class	5
isoclinism	isoclinic class of #93782
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂×C₄⋊C₈	C₂×C₄
Frattini, cosocle	#1984	C₂²
prime radical	C₂³	#327
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1984 ◁ G
LECS	G ▷ #1984 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

178 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=fhi,d^2=i,e^2=h,f^2=hi,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,d^b=defh,e^b=eh,f^b=fhi,g^b=gi,d^c=dgh,e^c=ei,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 997);