#265119

order	512
nilpotency class	3
isoclinism	isoclinic class of #265099
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	C₂×#3354	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

869 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=e,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,f^c=fi,g^c=ghi

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, 172315);