#480370

order	512
nilpotency class	4
isoclinism	isoclinic class of #480319
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×#1466
prime radical	C₂×#525	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #2873 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

910 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292716, #72292724, C₂×#480370
maximal normal subgroup of	#72292716, #72292724, C₂×#480370
maximal quotient of	C₂×#480370, #72292827

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387566);