#230074

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	C₂×AGL(1,ℤ/8)	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=di,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137270);