#481546

order	512
nilpotency class	4
isoclinism	isoclinic class of #481477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#36032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

947 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,d^c=di,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388742);