#277925

order	512
nilpotency class	3
isoclinism	isoclinic class of #277891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=i,d^2=ei,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185121);