#495946

order	512
nilpotency class	4
isoclinism	isoclinic class of #483873
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,17)(11,19)(12,24,26,37)(14,29,28,18)(15,30)(16,34,36,27)(20,38)(21,33)(22,35)(23,39)(25,31)(32,40);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,25)(13,23)(14,20)(16,35)(17,33)(18,31)(19,37)(21,34)(22,28)(24,39)(26,40)(27,30)(29,32)(36,38);(11,20,23,22)(12,26)(15,31,33,32)(16,36)(19,38,39,35)(21,40,30,25)(24,37)(27,34)

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=gi,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265093, #72265106, #72265140, …
maximal normal subgroup of	#72265093, #72265106, #72265140, …
maximal quotient of	#72343758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403142);