#482814

order	512
nilpotency class	4
isoclinism	isoclinic class of #482734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#553 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

869 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,d^c=dhi,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, 390010);