#481680

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#2617	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1127 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)(11,19)(12,23,25,16)(13,26,27,17)(14,18)(15,22,29,36)(20,33,34,31)(21,32)(24,30)(28,35)(37,40)(38,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,28)(17,19)(18,29)(23,37)(24,35)(25,34)(26,32)(27,36)(30,31)(33,38)(39,40);(1,3,5,4)(2,6,7,8)(9,12,14,25)(10,16,18,23)(11,20,21,34)(13,24,27,39)(15,28,29,37)(17,30,26,38)(19,31,32,33)(22,35,36,40);(11,21)(15,29)(19,32)(20,34)(22,36)(28,37)(31,33)(35,40)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,e^2=h,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	#72340394, #72340403, #72340407, …
maximal normal subgroup of	#72340394, #72340403, #72340407, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388876);