#93854

order	512
nilpotency class	5
isoclinism	isoclinic class of #93812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1345
commutator	C₂×C₄⋊C₈	C₂×C₄
Frattini, cosocle	#1842	C₂²
prime radical	C₂³	#327
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1842 ◁ G
LECS	G ▷ #1842 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 162 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) = 64

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

Polycyclic presentation:

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

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, 1050);