#351509

order	512
nilpotency class	3
isoclinism	isoclinic class of #348398
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

604 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)(9,10,14,11)(12,13,16,15)(17,18)(19,26)(20,23)(21,24)(22,27)(25,29)(28,31)(30,32);(1,3,4,7)(2,5,6,8)(9,11,13,16,14,10,15,12)(17,19)(18,22)(20,27)(21,28)(23,26)(24,30)(25,31)(29,32);(2,6)(5,8)(10,11)(12,16)(18,23)(19,27)(24,25)(28,32);(9,12,15,10,14,16,13,11)(18,24)(21,29)(22,30)(23,25)(26,31)(28,32)

Polycyclic presentation:

2^9|a^2=h,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=ci,d^c=dh,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, 258705);