#482830

order	512
nilpotency class	4
isoclinism	isoclinic class of #482734
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×C₄⋊C₄	C₂⁴
prime radical	#40607	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#559 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

870 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,e^2=h,f^2=i,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, 390026);