#348624

order	512
nilpotency class	3
isoclinism	isoclinic class of #348602
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3483	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

822 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255820);