#402040

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=i,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^d=hi

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, 309236);