#481702

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
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	#2617	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1087 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)(9,10,13,18,14,17,27,32)(11,19,22,33,21,15,28,30)(12,23,24,34,25,16,26,31)(20,29,37,39,36,35,38,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,29)(17,19)(18,33)(23,35)(24,37)(25,36)(26,38)(27,28)(30,32)(31,40)(34,39);(1,3,5,4)(2,6,7,8)(9,12,14,25)(10,16,17,23)(11,20,21,36)(13,26,27,24)(15,29,19,35)(18,34,32,31)(22,38,28,37)(30,40,33,39);(11,21)(15,19)(20,36)(22,28)(29,35)(30,33)(37,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340322, #72340403, #72340405, …
maximal normal subgroup of	#72340322, #72340403, #72340405, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388898);