#496004

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	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

548 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)(3,8)(4,6)(5,7)(9,10,12,15)(11,19)(13,26,24,38)(14,29,25,18)(16,32)(17,35,31,28)(20,39)(21,37)(22,30)(23,36)(27,33)(34,40);(1,3,5,4)(2,6,7,8)(10,15)(16,30)(17,31)(18,29)(19,37)(26,38)(33,34)(36,39);(9,11)(10,16)(12,21)(13,27)(14,20)(15,30)(17,36)(18,33)(19,38)(22,35)(23,25)(24,40)(26,37)(28,32)(29,34)(31,39);(1,4,5,3)(2,6,7,8)(11,20,21,23)(13,24)(16,33,30,34)(17,31)(19,39,37,36)(22,40,32,27)(26,38)(28,35)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=eg,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	#72309480, #72309671, #72310200, …
maximal normal subgroup of	#72309480, #72309671, #72310200, …
maximal quotient of	#72343497, #72343596

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403200);