#120991

order	512
nilpotency class	3
isoclinism	isoclinic class of #120987
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

438 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28187);