#6445482

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,19)(17,20);(2,4)(14,18)(15,16)(19,20);(5,7,8,12)(6,9,10,11)(13,14)(15,18)(16,19)(17,20);(13,14)(15,18)(16,19)(17,20);(13,15,17,16)(14,18,20,19)

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305395, C₂²⋊Q₈×AGL(1,ℤ/8), #72305413
maximal normal subgroup of	#72305395, C₂²⋊Q₈×AGL(1,ℤ/8), #72305413
maximal quotient of	#72305740

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352678);