#230068

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
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

576 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:

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

Polycyclic presentation:

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

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, 137264);