#127760

order	512
nilpotency class	4
isoclinism	isoclinic class of #127645
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=ch,d^b=dh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310095, #72310106, #72310109, …
maximal normal subgroup of	#72310095, #72310106, #72310109, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34956);