#496838

order	512
nilpotency class	4
isoclinism	isoclinic class of #483919
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	#33519	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

716 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)(3,7)(4,6)(5,8)(9,10,13,18,15,21,29,20)(11,16,14,30,23,19,32,34)(12,24,27,39,28,35,26,33)(17,31,37,22,38,25,36,40);(1,3)(2,5)(4,8)(6,7)(9,11,13,14,15,23,29,32)(10,16,18,30,21,19,20,34)(12,25,27,40,28,31,26,22)(17,35,37,33,38,24,36,39);(3,8)(5,7)(9,12)(10,17)(11,22)(13,26)(14,31)(15,28)(16,33)(18,36)(19,39)(20,37)(21,38)(23,40)(24,34)(25,32)(27,29)(30,35);(2,6)(5,7)(9,13,15,29)(10,18,21,20)(11,14,23,32)(12,26,28,27)(16,30,19,34)(17,36,38,37)(22,31,40,25)(24,33,35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265445, #72265470, #72265504, …
maximal normal subgroup of	#72265445, #72265470, #72265504, …
maximal quotient of	#72343541

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404034);