#480584

order	512
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

939 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=chi,f^b=fg,f^e=fi

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, 387780);