#496066

order	512
nilpotency class	4
isoclinism	isoclinic class of #483956
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

565 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309480, #72310194, #72310208, …
maximal normal subgroup of	#72309480, #72310194, #72310208, …
maximal quotient of	#72343502, #72343588

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403262);