#351401

order	512
nilpotency class	3
isoclinism	isoclinic class of #348358
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

623 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)(3,5)(4,6)(7,8)(9,10,11,13)(12,14,16,15)(18,22)(19,26)(20,23)(24,28)(25,32)(29,31);(1,3,4,7)(2,5,6,8)(10,13)(14,15)(17,18,23,28,21,24,20,22)(19,25,30,32,26,31,27,29);(2,6)(5,8)(10,13)(14,15)(18,24)(22,28)(25,31)(29,32);(10,14)(12,16)(13,15)(17,19,20,27,21,26,23,30)(18,25,22,29,24,31,28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=efh,d^2=ei,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ch,g^d=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258597);