#317969

order	512
nilpotency class	3
isoclinism	isoclinic class of #317966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,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, 225165);