#251118

order	512
nilpotency class	3
isoclinism	isoclinic class of #251031
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

999 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158314);