#118012

order	512
nilpotency class	3
isoclinism	isoclinic class of #9996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

731 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:

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

Polycyclic presentation:

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

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, 25208);