#99904

order	512
nilpotency class	3
isoclinism	isoclinic class of #99899
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

477 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=egh,f^c=fgh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7100);