#200039

order	512
nilpotency class	3
isoclinism	isoclinic class of #199797
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1199 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107235);