#402280

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

1044 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=i,c^2=fi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,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, 309476);