#482940

order	512
nilpotency class	4
isoclinism	isoclinic class of #482716
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	#2408	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#556 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

909 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,22,28,23,27,29,31)(19,24,20,25,30,32,21,26);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19)(18,24,27,32)(20,22)(21,29)(23,30)(25,28,26,31);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,20,23,21)(18,25,27,26)(19,22,30,29)(24,28,32,31);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(17,21,23,20)(18,26,27,25)(19,29,30,22)(24,31,32,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=be,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,d^c=dhi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390136);