#230082

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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

561 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:

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

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=dh,f^a=fh,d^b=dei,e^b=eh,g^b=gh,d^c=di,f^c=fh

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, 137278);