#205525

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	#60782	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1020 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=f,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,g^b=gi,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112721);