#121947

order	512
nilpotency class	3
isoclinism	isoclinic class of #121940
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi,g^c=ghi

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, 29143);