#481039

order	512
nilpotency class	4
isoclinism	isoclinic class of #480866
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	#39128	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#553 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388235);