#480318

order	512
nilpotency class	4
isoclinism	isoclinic class of #480314
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1464
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=i,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339826, #72339830, #72339831
maximal normal subgroup of	#72339826, #72339830, #72339831

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387514);