#98633

order	512
nilpotency class	3
isoclinism	isoclinic class of #98566
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=g,d^2=h,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=eg,f^c=fgi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5829);