#319697

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

891 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)(3,7)(4,6)(5,8)(17,18,23,29,22,20,21,25)(19,24,30,32,27,26,28,31);(1,3,4,8)(2,5,6,7)(9,10,11,14,12,15,16,13)(17,19,22,27)(18,24,20,26)(21,28,23,30)(25,31,29,32);(10,13)(11,16)(14,15)(17,20,22,18)(19,26,27,24)(21,25,23,29)(28,31,30,32);(2,6)(5,7)(19,27)(24,26)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=egh,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226893);