#265216

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

844 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172412);