#480763

order	512
nilpotency class	4
isoclinism	isoclinic class of #480656
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

919 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,e^b=ei,f^b=fg,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387959);