#449961

order	512
nilpotency class	3
isoclinism	isoclinic class of #449961
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15954
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1087 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 357157);