#383381

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1432 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)(3,4)(9,10)(11,13)(12,14)(15,16)(17,18,21,24,22,26,20,25)(19,23,28,30,29,32,27,31);(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19,22,29)(18,23,26,32)(20,27,21,28)(24,30,25,31);(2,4)(6,8)(9,12,13,16)(10,14,11,15)(18,24)(20,21)(23,30)(25,26)(27,28)(31,32);(2,4)(10,11)(14,15)(18,25)(20,21)(23,31)(24,26)(27,28)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290577);