#481997

order	512
nilpotency class	4
isoclinism	isoclinic class of #481837
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#35692	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1121 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)(3,9)(4,16,18,7)(5,8)(6,12)(10,17,21,30)(11,19)(13,25)(14,28,26,20)(15,22)(23,27)(24,29,31,32);(2,7)(4,17)(8,10)(9,20)(11,21)(12,16)(14,29)(18,30)(22,24)(23,31)(25,28)(26,32);(2,8)(3,13)(7,10)(9,23)(11,12)(14,26)(15,27)(16,21)(20,31)(22,25)(24,28)(29,32);(1,3,5,15)(2,9,8,22)(4,14,17,29)(6,13,19,27)(7,20,10,24)(11,23,12,25)(16,28,21,31)(18,26,30,32)

Polycyclic presentation:

2^9|a^2=f,d^2=h,e^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389193);