#94525

order	512
nilpotency class	6
isoclinism	isoclinic class of #94523
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7338
commutator	C₂²×Q₈	C₂×C₈
Frattini, cosocle	#1784	C₂²
prime radical	#3486	C₄
LCS	G ▷ C₂²×Q₈ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1784 ◁ G
LECS	G ▷ #1784 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 285 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=i,d^2=f,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=ei,f^b=fhi,g^b=gi,d^c=dgi,e^c=ei,f^c=fi,e^d=eh,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, 1721);