#218823

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

605 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)(3,7)(4,6)(5,8)(9,10,14,11,13,15,12,16)(18,23)(19,26)(20,22)(24,31)(27,32)(28,30);(1,3)(2,5)(4,8)(6,7)(9,11,14,15,13,16,12,10)(17,18,21,23)(19,27,28,24)(20,29,22,25)(26,31,30,32);(3,8)(5,7)(9,12,13,14)(10,11,15,16)(17,19,20,30,21,28,22,26)(18,24,29,31,23,27,25,32);(2,6)(5,7)(10,15)(11,16)(17,20,21,22)(18,25,23,29)(19,26,28,30)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,d^c=dh

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, 126019);