#480896

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	#37832	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#546 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,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, 388092);